diff --git a/Lib/unittest/suite.py b/Lib/unittest/suite.py
index 41993f9cf69..6f45b6fe5f6 100644
--- a/Lib/unittest/suite.py
+++ b/Lib/unittest/suite.py
@@ -149,6 +149,7 @@ class TestSuite(BaseTestSuite):
         if getattr(currentClass, "__unittest_skip__", False):
             return
 
+        failed = False
         try:
             currentClass._classSetupFailed = False
         except TypeError:
@@ -157,27 +158,32 @@ class TestSuite(BaseTestSuite):
             pass
 
         setUpClass = getattr(currentClass, 'setUpClass', None)
+        doClassCleanups = getattr(currentClass, 'doClassCleanups', None)
         if setUpClass is not None:
             _call_if_exists(result, '_setupStdout')
             try:
-                setUpClass()
-            except Exception as e:
-                if isinstance(result, _DebugResult):
-                    raise
-                currentClass._classSetupFailed = True
-                className = util.strclass(currentClass)
-                self._createClassOrModuleLevelException(result, e,
-                                                        'setUpClass',
-                                                        className)
+                try:
+                    setUpClass()
+                except Exception as e:
+                    if isinstance(result, _DebugResult):
+                        raise
+                    failed = True
+                    try:
+                        currentClass._classSetupFailed = True
+                    except TypeError:
+                        pass
+                    className = util.strclass(currentClass)
+                    self._createClassOrModuleLevelException(result, e,
+                                                            'setUpClass',
+                                                            className)
+                if failed and doClassCleanups is not None:
+                    doClassCleanups()
+                    for exc_info in currentClass.tearDown_exceptions:
+                        self._createClassOrModuleLevelException(
+                                result, exc_info[1], 'setUpClass', className,
+                                info=exc_info)
             finally:
                 _call_if_exists(result, '_restoreStdout')
-                if currentClass._classSetupFailed is True:
-                    currentClass.doClassCleanups()
-                    if len(currentClass.tearDown_exceptions) > 0:
-                        for exc in currentClass.tearDown_exceptions:
-                            self._createClassOrModuleLevelException(
-                                    result, exc[1], 'setUpClass', className,
-                                    info=exc)
 
     def _get_previous_module(self, result):
         previousModule = None
@@ -205,20 +211,22 @@ class TestSuite(BaseTestSuite):
         if setUpModule is not None:
             _call_if_exists(result, '_setupStdout')
             try:
-                setUpModule()
-            except Exception as e:
                 try:
-                    case.doModuleCleanups()
-                except Exception as exc:
-                    self._createClassOrModuleLevelException(result, exc,
+                    setUpModule()
+                except Exception as e:
+                    if isinstance(result, _DebugResult):
+                        raise
+                    result._moduleSetUpFailed = True
+                    self._createClassOrModuleLevelException(result, e,
                                                             'setUpModule',
                                                             currentModule)
-                if isinstance(result, _DebugResult):
-                    raise
-                result._moduleSetUpFailed = True
-                self._createClassOrModuleLevelException(result, e,
-                                                        'setUpModule',
-                                                        currentModule)
+                if result._moduleSetUpFailed:
+                    try:
+                        case.doModuleCleanups()
+                    except Exception as e:
+                        self._createClassOrModuleLevelException(result, e,
+                                                                'setUpModule',
+                                                                currentModule)
             finally:
                 _call_if_exists(result, '_restoreStdout')
 
@@ -251,30 +259,33 @@ class TestSuite(BaseTestSuite):
         except KeyError:
             return
 
-        tearDownModule = getattr(module, 'tearDownModule', None)
-        if tearDownModule is not None:
-            _call_if_exists(result, '_setupStdout')
+        _call_if_exists(result, '_setupStdout')
+        try:
+            tearDownModule = getattr(module, 'tearDownModule', None)
+            if tearDownModule is not None:
+                try:
+                    tearDownModule()
+                except Exception as e:
+                    if isinstance(result, _DebugResult):
+                        raise
+                    self._createClassOrModuleLevelException(result, e,
+                                                            'tearDownModule',
+                                                            previousModule)
             try:
-                tearDownModule()
+                case.doModuleCleanups()
             except Exception as e:
                 if isinstance(result, _DebugResult):
                     raise
                 self._createClassOrModuleLevelException(result, e,
                                                         'tearDownModule',
                                                         previousModule)
-            finally:
-                _call_if_exists(result, '_restoreStdout')
-                try:
-                    case.doModuleCleanups()
-                except Exception as e:
-                    self._createClassOrModuleLevelException(result, e,
-                                                            'tearDownModule',
-                                                            previousModule)
+        finally:
+            _call_if_exists(result, '_restoreStdout')
 
     def _tearDownPreviousClass(self, test, result):
         previousClass = getattr(result, '_previousTestClass', None)
         currentClass = test.__class__
-        if currentClass == previousClass:
+        if currentClass == previousClass or previousClass is None:
             return
         if getattr(previousClass, '_classSetupFailed', False):
             return
@@ -284,27 +295,34 @@ class TestSuite(BaseTestSuite):
             return
 
         tearDownClass = getattr(previousClass, 'tearDownClass', None)
-        if tearDownClass is not None:
-            _call_if_exists(result, '_setupStdout')
-            try:
-                tearDownClass()
-            except Exception as e:
-                if isinstance(result, _DebugResult):
-                    raise
-                className = util.strclass(previousClass)
-                self._createClassOrModuleLevelException(result, e,
-                                                        'tearDownClass',
-                                                        className)
-            finally:
-                _call_if_exists(result, '_restoreStdout')
-                previousClass.doClassCleanups()
-                if len(previousClass.tearDown_exceptions) > 0:
-                    for exc in previousClass.tearDown_exceptions:
-                        className = util.strclass(previousClass)
-                        self._createClassOrModuleLevelException(result, exc[1],
-                                                                'tearDownClass',
-                                                                className,
-                                                                info=exc)
+        doClassCleanups = getattr(previousClass, 'doClassCleanups', None)
+        if tearDownClass is None and doClassCleanups is None:
+            return
+
+        _call_if_exists(result, '_setupStdout')
+        try:
+            if tearDownClass is not None:
+                try:
+                    tearDownClass()
+                except Exception as e:
+                    if isinstance(result, _DebugResult):
+                        raise
+                    className = util.strclass(previousClass)
+                    self._createClassOrModuleLevelException(result, e,
+                                                            'tearDownClass',
+                                                            className)
+            if doClassCleanups is not None:
+                doClassCleanups()
+                for exc_info in previousClass.tearDown_exceptions:
+                    if isinstance(result, _DebugResult):
+                        raise exc_info[1]
+                    className = util.strclass(previousClass)
+                    self._createClassOrModuleLevelException(result, exc_info[1],
+                                                            'tearDownClass',
+                                                            className,
+                                                            info=exc_info)
+        finally:
+            _call_if_exists(result, '_restoreStdout')
 
 
 class _ErrorHolder(object):
diff --git a/Lib/unittest/test/test_result.py b/Lib/unittest/test/test_result.py
index a4af67bd8d5..d6efc7ef066 100644
--- a/Lib/unittest/test/test_result.py
+++ b/Lib/unittest/test/test_result.py
@@ -2,10 +2,11 @@ import io
 import sys
 import textwrap
 
-from test.support import warnings_helper
+from test.support import warnings_helper, captured_stdout, captured_stderr
 
 import traceback
 import unittest
+from unittest.util import strclass
 
 
 class MockTraceback(object):
@@ -22,6 +23,16 @@ def restore_traceback():
     unittest.result.traceback = traceback
 
 
+def bad_cleanup1():
+    print('do cleanup1')
+    raise TypeError('bad cleanup1')
+
+
+def bad_cleanup2():
+    print('do cleanup2')
+    raise ValueError('bad cleanup2')
+
+
 class Test_TestResult(unittest.TestCase):
     # Note: there are not separate tests for TestResult.wasSuccessful(),
     # TestResult.errors, TestResult.failures, TestResult.testsRun or
@@ -633,36 +644,320 @@ class TestOutputBuffering(unittest.TestCase):
             self.assertEqual(result._original_stderr.getvalue(), expectedErrMessage)
             self.assertMultiLineEqual(message, expectedFullMessage)
 
+    def testBufferSetUp(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def setUp(self):
+                print('set up')
+                1/0
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\nset up\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 1)
+        description = f'test_foo ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferTearDown(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def tearDown(self):
+                print('tear down')
+                1/0
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\ntear down\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 1)
+        description = f'test_foo ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferDoCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def setUp(self):
+                print('set up')
+                self.addCleanup(bad_cleanup1)
+                self.addCleanup(bad_cleanup2)
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\nset up\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 2)
+        description = f'test_foo ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('TypeError: bad cleanup1', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferSetUp_DoCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def setUp(self):
+                print('set up')
+                self.addCleanup(bad_cleanup1)
+                self.addCleanup(bad_cleanup2)
+                1/0
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\nset up\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 3)
+        description = f'test_foo ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[2]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('TypeError: bad cleanup1', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferTearDown_DoCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def setUp(self):
+                print('set up')
+                self.addCleanup(bad_cleanup1)
+                self.addCleanup(bad_cleanup2)
+            def tearDown(self):
+                print('tear down')
+                1/0
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\nset up\ntear down\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 3)
+        description = f'test_foo ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[2]
+        self.assertEqual(str(test_case), description)
+        self.assertIn('TypeError: bad cleanup1', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
     def testBufferSetupClass(self):
-        result = unittest.TestResult()
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
         result.buffer = True
 
         class Foo(unittest.TestCase):
             @classmethod
             def setUpClass(cls):
+                print('set up class')
                 1/0
             def test_foo(self):
                 pass
         suite = unittest.TestSuite([Foo('test_foo')])
         suite(result)
+        expected_out = '\nStdout:\nset up class\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
         self.assertEqual(len(result.errors), 1)
+        description = f'setUpClass ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
 
     def testBufferTearDownClass(self):
-        result = unittest.TestResult()
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
         result.buffer = True
 
         class Foo(unittest.TestCase):
             @classmethod
             def tearDownClass(cls):
+                print('tear down class')
                 1/0
             def test_foo(self):
                 pass
         suite = unittest.TestSuite([Foo('test_foo')])
         suite(result)
+        expected_out = '\nStdout:\ntear down class\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
         self.assertEqual(len(result.errors), 1)
+        description = f'tearDownClass ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferDoClassCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                print('set up class')
+                cls.addClassCleanup(bad_cleanup1)
+                cls.addClassCleanup(bad_cleanup2)
+            @classmethod
+            def tearDownClass(cls):
+                print('tear down class')
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\ntear down class\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 2)
+        description = f'tearDownClass ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('TypeError: bad cleanup1', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferSetupClass_DoClassCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                print('set up class')
+                cls.addClassCleanup(bad_cleanup1)
+                cls.addClassCleanup(bad_cleanup2)
+                1/0
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\nset up class\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 3)
+        description = f'setUpClass ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn('\nStdout:\nset up class\n', formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[2]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('TypeError: bad cleanup1', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferTearDownClass_DoClassCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                print('set up class')
+                cls.addClassCleanup(bad_cleanup1)
+                cls.addClassCleanup(bad_cleanup2)
+            @classmethod
+            def tearDownClass(cls):
+                print('tear down class')
+                1/0
+            def test_foo(self):
+                pass
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\ntear down class\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 3)
+        description = f'tearDownClass ({strclass(Foo)})'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn('\nStdout:\ntear down class\n', formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+        test_case, formatted_exc = result.errors[2]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('TypeError: bad cleanup1', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
 
     def testBufferSetUpModule(self):
-        result = unittest.TestResult()
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
         result.buffer = True
 
         class Foo(unittest.TestCase):
@@ -671,6 +966,7 @@ class TestOutputBuffering(unittest.TestCase):
         class Module(object):
             @staticmethod
             def setUpModule():
+                print('set up module')
                 1/0
 
         Foo.__module__ = 'Module'
@@ -678,10 +974,18 @@ class TestOutputBuffering(unittest.TestCase):
         self.addCleanup(sys.modules.pop, 'Module')
         suite = unittest.TestSuite([Foo('test_foo')])
         suite(result)
+        expected_out = '\nStdout:\nset up module\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
         self.assertEqual(len(result.errors), 1)
+        description = 'setUpModule (Module)'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
 
     def testBufferTearDownModule(self):
-        result = unittest.TestResult()
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
         result.buffer = True
 
         class Foo(unittest.TestCase):
@@ -690,6 +994,7 @@ class TestOutputBuffering(unittest.TestCase):
         class Module(object):
             @staticmethod
             def tearDownModule():
+                print('tear down module')
                 1/0
 
         Foo.__module__ = 'Module'
@@ -697,7 +1002,124 @@ class TestOutputBuffering(unittest.TestCase):
         self.addCleanup(sys.modules.pop, 'Module')
         suite = unittest.TestSuite([Foo('test_foo')])
         suite(result)
+        expected_out = '\nStdout:\ntear down module\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
         self.assertEqual(len(result.errors), 1)
+        description = 'tearDownModule (Module)'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferDoModuleCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def test_foo(self):
+                pass
+        class Module(object):
+            @staticmethod
+            def setUpModule():
+                print('set up module')
+                unittest.addModuleCleanup(bad_cleanup1)
+                unittest.addModuleCleanup(bad_cleanup2)
+
+        Foo.__module__ = 'Module'
+        sys.modules['Module'] = Module
+        self.addCleanup(sys.modules.pop, 'Module')
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 1)
+        description = 'tearDownModule (Module)'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferSetUpModule_DoModuleCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def test_foo(self):
+                pass
+        class Module(object):
+            @staticmethod
+            def setUpModule():
+                print('set up module')
+                unittest.addModuleCleanup(bad_cleanup1)
+                unittest.addModuleCleanup(bad_cleanup2)
+                1/0
+
+        Foo.__module__ = 'Module'
+        sys.modules['Module'] = Module
+        self.addCleanup(sys.modules.pop, 'Module')
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\nset up module\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 2)
+        description = 'setUpModule (Module)'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn('\nStdout:\nset up module\n', formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertIn(expected_out, formatted_exc)
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
+
+    def testBufferTearDownModule_DoModuleCleanups(self):
+        with captured_stdout() as stdout:
+            result = unittest.TestResult()
+        result.buffer = True
+
+        class Foo(unittest.TestCase):
+            def test_foo(self):
+                pass
+        class Module(object):
+            @staticmethod
+            def setUpModule():
+                print('set up module')
+                unittest.addModuleCleanup(bad_cleanup1)
+                unittest.addModuleCleanup(bad_cleanup2)
+            @staticmethod
+            def tearDownModule():
+                print('tear down module')
+                1/0
+
+        Foo.__module__ = 'Module'
+        sys.modules['Module'] = Module
+        self.addCleanup(sys.modules.pop, 'Module')
+        suite = unittest.TestSuite([Foo('test_foo')])
+        suite(result)
+        expected_out = '\nStdout:\ntear down module\ndo cleanup2\ndo cleanup1\n'
+        self.assertEqual(stdout.getvalue(), expected_out)
+        self.assertEqual(len(result.errors), 2)
+        description = 'tearDownModule (Module)'
+        test_case, formatted_exc = result.errors[0]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ZeroDivisionError: division by zero', formatted_exc)
+        self.assertNotIn('ValueError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn('\nStdout:\ntear down module\n', formatted_exc)
+        test_case, formatted_exc = result.errors[1]
+        self.assertEqual(test_case.description, description)
+        self.assertIn('ValueError: bad cleanup2', formatted_exc)
+        self.assertNotIn('ZeroDivisionError', formatted_exc)
+        self.assertNotIn('TypeError', formatted_exc)
+        self.assertIn(expected_out, formatted_exc)
 
 
 if __name__ == '__main__':
diff --git a/Lib/unittest/test/test_runner.py b/Lib/unittest/test/test_runner.py
index dd9a1b6d9ae..453e6c3d11c 100644
--- a/Lib/unittest/test/test_runner.py
+++ b/Lib/unittest/test/test_runner.py
@@ -222,14 +222,42 @@ class TestClassCleanup(unittest.TestCase):
         self.assertEqual(ordering,
                          ['setUpClass', 'test', 'tearDownClass', 'cleanup_good'])
 
-    def test_debug_executes_classCleanUp(self):
+    def test_run_class_cleanUp_without_tearDownClass(self):
         ordering = []
+        blowUp = True
 
         class TestableTest(unittest.TestCase):
             @classmethod
             def setUpClass(cls):
                 ordering.append('setUpClass')
                 cls.addClassCleanup(cleanup, ordering)
+                if blowUp:
+                    raise Exception()
+            def testNothing(self):
+                ordering.append('test')
+            @classmethod
+            @property
+            def tearDownClass(cls):
+                raise AttributeError
+
+        runTests(TestableTest)
+        self.assertEqual(ordering, ['setUpClass', 'cleanup_good'])
+
+        ordering = []
+        blowUp = False
+        runTests(TestableTest)
+        self.assertEqual(ordering,
+                         ['setUpClass', 'test', 'cleanup_good'])
+
+    def test_debug_executes_classCleanUp(self):
+        ordering = []
+        blowUp = False
+
+        class TestableTest(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                ordering.append('setUpClass')
+                cls.addClassCleanup(cleanup, ordering, blowUp=blowUp)
             def testNothing(self):
                 ordering.append('test')
             @classmethod
@@ -241,6 +269,48 @@ class TestClassCleanup(unittest.TestCase):
         self.assertEqual(ordering,
                          ['setUpClass', 'test', 'tearDownClass', 'cleanup_good'])
 
+        ordering = []
+        blowUp = True
+        suite = unittest.defaultTestLoader.loadTestsFromTestCase(TestableTest)
+        with self.assertRaises(Exception) as cm:
+            suite.debug()
+        self.assertEqual(str(cm.exception), 'CleanUpExc')
+        self.assertEqual(ordering,
+                         ['setUpClass', 'test', 'tearDownClass', 'cleanup_exc'])
+
+    def test_debug_executes_classCleanUp_when_teardown_exception(self):
+        ordering = []
+        blowUp = False
+
+        class TestableTest(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                ordering.append('setUpClass')
+                cls.addClassCleanup(cleanup, ordering, blowUp=blowUp)
+            def testNothing(self):
+                ordering.append('test')
+            @classmethod
+            def tearDownClass(cls):
+                raise Exception('TearDownClassExc')
+
+        suite = unittest.defaultTestLoader.loadTestsFromTestCase(TestableTest)
+        with self.assertRaises(Exception) as cm:
+            suite.debug()
+        self.assertEqual(str(cm.exception), 'TearDownClassExc')
+        self.assertEqual(ordering, ['setUpClass', 'test'])
+        self.assertTrue(TestableTest._class_cleanups)
+        TestableTest._class_cleanups.clear()
+
+        ordering = []
+        blowUp = True
+        suite = unittest.defaultTestLoader.loadTestsFromTestCase(TestableTest)
+        with self.assertRaises(Exception) as cm:
+            suite.debug()
+        self.assertEqual(str(cm.exception), 'TearDownClassExc')
+        self.assertEqual(ordering, ['setUpClass', 'test'])
+        self.assertTrue(TestableTest._class_cleanups)
+        TestableTest._class_cleanups.clear()
+
     def test_doClassCleanups_with_errors_addClassCleanUp(self):
         class TestableTest(unittest.TestCase):
             def testNothing(self):
@@ -332,6 +402,7 @@ class TestClassCleanup(unittest.TestCase):
         self.assertEqual(ordering,
                          ['setUpClass', 'setUp', 'test',
                           'tearDownClass', 'cleanup_exc'])
+
         ordering = []
         class_blow_up = True
         method_blow_up = False
@@ -355,6 +426,26 @@ class TestClassCleanup(unittest.TestCase):
                          ['setUpClass', 'setUp', 'tearDownClass',
                           'cleanup_exc'])
 
+    def test_with_errors_in_tearDownClass(self):
+        ordering = []
+        class TestableTest(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                ordering.append('setUpClass')
+                cls.addClassCleanup(cleanup, ordering)
+            def testNothing(self):
+                ordering.append('test')
+            @classmethod
+            def tearDownClass(cls):
+                ordering.append('tearDownClass')
+                raise Exception('TearDownExc')
+
+        result = runTests(TestableTest)
+        self.assertEqual(result.errors[0][1].splitlines()[-1],
+                         'Exception: TearDownExc')
+        self.assertEqual(ordering,
+                         ['setUpClass', 'test', 'tearDownClass', 'cleanup_good'])
+
 
 class TestModuleCleanUp(unittest.TestCase):
     def test_add_and_do_ModuleCleanup(self):
@@ -532,13 +623,69 @@ class TestModuleCleanUp(unittest.TestCase):
                           'tearDownModule2', 'cleanup_good'])
         self.assertEqual(unittest.case._module_cleanups, [])
 
-    def test_debug_module_executes_cleanUp(self):
+    def test_run_module_cleanUp_without_teardown(self):
         ordering = []
         class Module(object):
             @staticmethod
             def setUpModule():
                 ordering.append('setUpModule')
                 unittest.addModuleCleanup(cleanup, ordering)
+
+        class TestableTest(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                ordering.append('setUpClass')
+            def testNothing(self):
+                ordering.append('test')
+            @classmethod
+            def tearDownClass(cls):
+                ordering.append('tearDownClass')
+
+        TestableTest.__module__ = 'Module'
+        sys.modules['Module'] = Module
+        runTests(TestableTest)
+        self.assertEqual(ordering, ['setUpModule', 'setUpClass', 'test',
+                                    'tearDownClass', 'cleanup_good'])
+        self.assertEqual(unittest.case._module_cleanups, [])
+
+    def test_run_module_cleanUp_when_teardown_exception(self):
+        ordering = []
+        class Module(object):
+            @staticmethod
+            def setUpModule():
+                ordering.append('setUpModule')
+                unittest.addModuleCleanup(cleanup, ordering)
+            @staticmethod
+            def tearDownModule():
+                raise Exception('CleanUpExc')
+
+        class TestableTest(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                ordering.append('setUpClass')
+            def testNothing(self):
+                ordering.append('test')
+            @classmethod
+            def tearDownClass(cls):
+                ordering.append('tearDownClass')
+
+        TestableTest.__module__ = 'Module'
+        sys.modules['Module'] = Module
+        result = runTests(TestableTest)
+        self.assertEqual(result.errors[0][1].splitlines()[-1],
+                         'Exception: CleanUpExc')
+        self.assertEqual(ordering, ['setUpModule', 'setUpClass', 'test',
+                                    'tearDownClass', 'cleanup_good'])
+        self.assertEqual(unittest.case._module_cleanups, [])
+
+    def test_debug_module_executes_cleanUp(self):
+        ordering = []
+        blowUp = False
+        class Module(object):
+            @staticmethod
+            def setUpModule():
+                ordering.append('setUpModule')
+                unittest.addModuleCleanup(cleanup, ordering, blowUp=blowUp)
             @staticmethod
             def tearDownModule():
                 ordering.append('tearDownModule')
@@ -562,6 +709,60 @@ class TestModuleCleanUp(unittest.TestCase):
                           'tearDownModule', 'cleanup_good'])
         self.assertEqual(unittest.case._module_cleanups, [])
 
+        ordering = []
+        blowUp = True
+        suite = unittest.defaultTestLoader.loadTestsFromTestCase(TestableTest)
+        with self.assertRaises(Exception) as cm:
+            suite.debug()
+        self.assertEqual(str(cm.exception), 'CleanUpExc')
+        self.assertEqual(ordering, ['setUpModule', 'setUpClass', 'test',
+                                    'tearDownClass', 'tearDownModule', 'cleanup_exc'])
+        self.assertEqual(unittest.case._module_cleanups, [])
+
+    def test_debug_module_cleanUp_when_teardown_exception(self):
+        ordering = []
+        blowUp = False
+        class Module(object):
+            @staticmethod
+            def setUpModule():
+                ordering.append('setUpModule')
+                unittest.addModuleCleanup(cleanup, ordering, blowUp=blowUp)
+            @staticmethod
+            def tearDownModule():
+                raise Exception('TearDownModuleExc')
+
+        class TestableTest(unittest.TestCase):
+            @classmethod
+            def setUpClass(cls):
+                ordering.append('setUpClass')
+            def testNothing(self):
+                ordering.append('test')
+            @classmethod
+            def tearDownClass(cls):
+                ordering.append('tearDownClass')
+
+        TestableTest.__module__ = 'Module'
+        sys.modules['Module'] = Module
+        suite = unittest.defaultTestLoader.loadTestsFromTestCase(TestableTest)
+        with self.assertRaises(Exception) as cm:
+            suite.debug()
+        self.assertEqual(str(cm.exception), 'TearDownModuleExc')
+        self.assertEqual(ordering, ['setUpModule', 'setUpClass', 'test',
+                                    'tearDownClass'])
+        self.assertTrue(unittest.case._module_cleanups)
+        unittest.case._module_cleanups.clear()
+
+        ordering = []
+        blowUp = True
+        suite = unittest.defaultTestLoader.loadTestsFromTestCase(TestableTest)
+        with self.assertRaises(Exception) as cm:
+            suite.debug()
+        self.assertEqual(str(cm.exception), 'TearDownModuleExc')
+        self.assertEqual(ordering, ['setUpModule', 'setUpClass', 'test',
+                                    'tearDownClass'])
+        self.assertTrue(unittest.case._module_cleanups)
+        unittest.case._module_cleanups.clear()
+
     def test_addClassCleanup_arg_errors(self):
         cleanups = []
         def cleanup(*args, **kwargs):
@@ -717,9 +918,9 @@ class TestModuleCleanUp(unittest.TestCase):
         method_blow_up = False
         result = runTests(TestableTest)
         self.assertEqual(result.errors[0][1].splitlines()[-1],
-                         'Exception: CleanUpExc')
-        self.assertEqual(result.errors[1][1].splitlines()[-1],
                          'Exception: ModuleExc')
+        self.assertEqual(result.errors[1][1].splitlines()[-1],
+                         'Exception: CleanUpExc')
         self.assertEqual(ordering, ['setUpModule', 'cleanup_exc'])
 
         ordering = []
diff --git a/Misc/NEWS.d/next/Library/2021-08-27-23-40-51.bpo-43913.Uo1Gt5.rst b/Misc/NEWS.d/next/Library/2021-08-27-23-40-51.bpo-43913.Uo1Gt5.rst
new file mode 100644
index 00000000000..cf3d5ee0e45
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2021-08-27-23-40-51.bpo-43913.Uo1Gt5.rst
@@ -0,0 +1,8 @@
+Fix bugs in cleaning up classes and modules in :mod:`unittest`:
+
+* Functions registered with :func:`~unittest.addModuleCleanup` were not called unless the user defines ``tearDownModule()`` in their test module.
+* Functions registered with :meth:`~unittest.TestCase.addClassCleanup` were not called if ``tearDownClass`` is set to ``None``.
+* Buffering in :class:`~unittest.TestResult` did not work with functions registered with ``addClassCleanup()`` and ``addModuleCleanup()``.
+* Errors in functions registered with ``addClassCleanup()`` and ``addModuleCleanup()`` were not handled correctly in buffered and debug modes.
+* Errors in ``setUpModule()`` and functions registered with ``addModuleCleanup()`` were reported in wrong order.
+* And several lesser bugs.