diff --git a/Lib/site.py b/Lib/site.py
index 0cc22eba2e5..a0489fa461c 100644
--- a/Lib/site.py
+++ b/Lib/site.py
@@ -55,6 +55,7 @@ ImportError exception, it is silently ignored.
 import sys
 import os
 import builtins
+import traceback
 
 # Prefixes for site-packages; add additional prefixes like /usr/local here
 PREFIXES = [sys.prefix, sys.exec_prefix]
@@ -146,17 +147,26 @@ def addpackage(sitedir, name, known_paths):
     except IOError:
         return
     with f:
-        for line in f:
+        for n, line in enumerate(f):
             if line.startswith("#"):
                 continue
-            if line.startswith(("import ", "import\t")):
-                exec(line)
-                continue
-            line = line.rstrip()
-            dir, dircase = makepath(sitedir, line)
-            if not dircase in known_paths and os.path.exists(dir):
-                sys.path.append(dir)
-                known_paths.add(dircase)
+            try:
+                if line.startswith(("import ", "import\t")):
+                    exec(line)
+                    continue
+                line = line.rstrip()
+                dir, dircase = makepath(sitedir, line)
+                if not dircase in known_paths and os.path.exists(dir):
+                    sys.path.append(dir)
+                    known_paths.add(dircase)
+            except Exception as err:
+                print("Error processing line {:d} of {}:\n".format(n+1, fullname),
+                      file=sys.stderr)
+                for record in traceback.format_exception(*sys.exc_info()):
+                    for line in record.splitlines():
+                        print('  '+line, file=sys.stderr)
+                print("\nRemainder of file ignored", file=sys.stderr)
+                break
     if reset:
         known_paths = None
     return known_paths
diff --git a/Lib/test/test_site.py b/Lib/test/test_site.py
index 7ed0ee24d48..343f8ef64be 100644
--- a/Lib/test/test_site.py
+++ b/Lib/test/test_site.py
@@ -6,6 +6,7 @@ executing have not been removed.
 """
 import unittest
 from test.support import run_unittest, TESTFN, EnvironmentVarGuard
+from test.support import captured_output
 import builtins
 import os
 import sys
@@ -83,6 +84,53 @@ class HelperFunctionsTests(unittest.TestCase):
         finally:
             pth_file.cleanup()
 
+    def make_pth(self, contents, pth_dir='.', pth_name=TESTFN):
+        # Create a .pth file and return its (abspath, basename).
+        pth_dir = os.path.abspath(pth_dir)
+        pth_basename = pth_name + '.pth'
+        pth_fn = os.path.join(pth_dir, pth_basename)
+        pth_file = open(pth_fn, 'w', encoding='utf-8')
+        self.addCleanup(lambda: os.remove(pth_fn))
+        pth_file.write(contents)
+        pth_file.close()
+        return pth_dir, pth_basename
+
+    def test_addpackage_import_bad_syntax(self):
+        # Issue 10642
+        pth_dir, pth_fn = self.make_pth("import bad)syntax\n")
+        with captured_output("stderr") as err_out:
+            site.addpackage(pth_dir, pth_fn, set())
+        self.assertRegexpMatches(err_out.getvalue(), "line 1")
+        self.assertRegexpMatches(err_out.getvalue(), os.path.join(pth_dir, pth_fn))
+        # XXX: the previous two should be independent checks so that the
+        # order doesn't matter.  The next three could be a single check
+        # but my regex foo isn't good enough to write it.
+        self.assertRegexpMatches(err_out.getvalue(), 'Traceback')
+        self.assertRegexpMatches(err_out.getvalue(), r'import bad\)syntax')
+        self.assertRegexpMatches(err_out.getvalue(), 'SyntaxError')
+
+    def test_addpackage_import_bad_exec(self):
+        # Issue 10642
+        pth_dir, pth_fn = self.make_pth("randompath\nimport nosuchmodule\n")
+        with captured_output("stderr") as err_out:
+            site.addpackage(pth_dir, pth_fn, set())
+        self.assertRegexpMatches(err_out.getvalue(), "line 2")
+        self.assertRegexpMatches(err_out.getvalue(), os.path.join(pth_dir, pth_fn))
+        # XXX: ditto previous XXX comment.
+        self.assertRegexpMatches(err_out.getvalue(), 'Traceback')
+        self.assertRegexpMatches(err_out.getvalue(), 'ImportError')
+
+    def test_addpackage_import_bad_pth_file(self):
+        # Issue 5258
+        pth_dir, pth_fn = self.make_pth("abc\x00def\n")
+        with captured_output("stderr") as err_out:
+            site.addpackage(pth_dir, pth_fn, set())
+        self.assertRegexpMatches(err_out.getvalue(), "line 1")
+        self.assertRegexpMatches(err_out.getvalue(), os.path.join(pth_dir, pth_fn))
+        # XXX: ditto previous XXX comment.
+        self.assertRegexpMatches(err_out.getvalue(), 'Traceback')
+        self.assertRegexpMatches(err_out.getvalue(), 'TypeError')
+
     def test_addsitedir(self):
         # Same tests for test_addpackage since addsitedir() essentially just
         # calls addpackage() for every .pth file in the directory
diff --git a/Misc/NEWS b/Misc/NEWS
index bd43ee770a0..d0dd3a19c60 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -24,6 +24,10 @@ Core and Builtins
 Library
 -------
 
+- Issue #5258/#10642: if site.py encounters a .pth file that generates an error,
+  it now prints the filename, line number, and traceback to stderr and skips
+  the rest of that individual file, instead of stopping processing entirely.
+
 - Issue #4871: The zipfile module now gives a more useful error message if
   an attempt is made to use a string to specify the archive password.