diff --git a/Lib/test/test_bigaddrspace.py b/Lib/test/test_bigaddrspace.py
index 3aedc4cf356..d383440356c 100644
--- a/Lib/test/test_bigaddrspace.py
+++ b/Lib/test/test_bigaddrspace.py
@@ -1,3 +1,13 @@
+"""
+These tests are meant to exercise that requests to create objects bigger
+than what the address space allows are properly met with an OverflowError
+(rather than crash weirdly).
+
+Primarily, this means 32-bit builds with at least 2 GB of available memory.
+You need to pass the -M option to regrtest (e.g. "-M 2.1G") for tests to
+be enabled.
+"""
+
 from test import support
 from test.support import bigaddrspacetest, MAX_Py_ssize_t
 
@@ -6,39 +16,66 @@ import operator
 import sys
 
 
-class StrTest(unittest.TestCase):
+class BytesTest(unittest.TestCase):
 
     @bigaddrspacetest
     def test_concat(self):
-        s1 = 'x' * MAX_Py_ssize_t
-        self.assertRaises(OverflowError, operator.add, s1, '?')
+        # Allocate a bytestring that's near the maximum size allowed by
+        # the address space, and then try to build a new, larger one through
+        # concatenation.
+        x = b"x" * (MAX_Py_ssize_t - 128)
+        self.assertRaises(OverflowError, operator.add, x, b"x" * 128)
 
     @bigaddrspacetest
     def test_optimized_concat(self):
-        x = 'x' * MAX_Py_ssize_t
-        try:
-            x = x + '?'     # this statement uses a fast path in ceval.c
-        except OverflowError:
-            pass
-        else:
-            self.fail("should have raised OverflowError")
-        try:
-            x += '?'        # this statement uses a fast path in ceval.c
-        except OverflowError:
-            pass
-        else:
-            self.fail("should have raised OverflowError")
-        self.assertEquals(len(x), MAX_Py_ssize_t)
+        x = b"x" * (MAX_Py_ssize_t - 128)
 
-    ### the following test is pending a patch
-    #   (http://mail.python.org/pipermail/python-dev/2006-July/067774.html)
-    #@bigaddrspacetest
-    #def test_repeat(self):
-    #    self.assertRaises(OverflowError, operator.mul, 'x', MAX_Py_ssize_t + 1)
+        with self.assertRaises(OverflowError) as cm:
+            # this statement uses a fast path in ceval.c
+            x = x + b"x" * 128
+
+        with self.assertRaises(OverflowError) as cm:
+            # this statement uses a fast path in ceval.c
+            x +=  b"x" * 128
+
+    @bigaddrspacetest
+    def test_repeat(self):
+        x = b"x" * (MAX_Py_ssize_t - 128)
+        self.assertRaises(OverflowError, operator.mul, x, 128)
+
+
+class StrTest(unittest.TestCase):
+
+    unicodesize = 2 if sys.maxunicode < 65536 else 4
+
+    @bigaddrspacetest
+    def test_concat(self):
+        # Create a string half the size that would fill the address space
+        x = "x" * (MAX_Py_ssize_t // (2 * self.unicodesize))
+        # Unicode objects trigger MemoryError in case an operation that's
+        # going to cause a size overflow is executed
+        self.assertRaises(MemoryError, operator.add, x, x)
+
+    @bigaddrspacetest
+    def test_optimized_concat(self):
+        x = "x" * (MAX_Py_ssize_t // (2 * self.unicodesize))
+
+        with self.assertRaises(MemoryError) as cm:
+            # this statement uses a fast path in ceval.c
+            x = x + x
+
+        with self.assertRaises(MemoryError) as cm:
+            # this statement uses a fast path in ceval.c
+            x +=  x
+
+    @bigaddrspacetest
+    def test_repeat(self):
+        x = "x" * (MAX_Py_ssize_t // (2 * self.unicodesize))
+        self.assertRaises(MemoryError, operator.mul, x, 2)
 
 
 def test_main():
-    support.run_unittest(StrTest)
+    support.run_unittest(BytesTest, StrTest)
 
 if __name__ == '__main__':
     if len(sys.argv) > 1:
diff --git a/Misc/NEWS b/Misc/NEWS
index 30eb8257ed5..05a0023e96d 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -702,6 +702,9 @@ Build
 Tests
 -----
 
+- Issue #3699: Fix test_bigaddrspace and extend it to test bytestrings
+  as well as unicode strings.  Initial patch by Sandro Tosi.
+
 - Issue #9628: fix runtests.sh -x option so more than one test can be excluded.
 
 - Issue #9894: Do not hardcode ENOENT in test_subprocess.