diff --git a/Lib/test/test_abc.py b/Lib/test/test_abc.py
index 86f31a9acb4..5ce57cc209e 100644
--- a/Lib/test/test_abc.py
+++ b/Lib/test/test_abc.py
@@ -448,15 +448,16 @@ class S(metaclass=abc_ABCMeta):
 
             # Also check that issubclass() propagates exceptions raised by
             # __subclasses__.
+            class CustomError(Exception): ...
             exc_msg = "exception from __subclasses__"
 
             def raise_exc():
-                raise Exception(exc_msg)
+                raise CustomError(exc_msg)
 
             class S(metaclass=abc_ABCMeta):
                 __subclasses__ = raise_exc
 
-            with self.assertRaisesRegex(Exception, exc_msg):
+            with self.assertRaisesRegex(CustomError, exc_msg):
                 issubclass(int, S)
 
         def test_subclasshook(self):
diff --git a/Lib/test/test_codecs.py b/Lib/test/test_codecs.py
index 376175f90f6..91d7eaf997a 100644
--- a/Lib/test/test_codecs.py
+++ b/Lib/test/test_codecs.py
@@ -2822,14 +2822,15 @@ def test_binary_to_text_denylists_text_transforms(self):
     def test_custom_zlib_error_is_noted(self):
         # Check zlib codec gives a good error for malformed input
         msg = "decoding with 'zlib_codec' codec failed"
-        with self.assertRaises(Exception) as failure:
+        with self.assertRaises(zlib.error) as failure:
             codecs.decode(b"hello", "zlib_codec")
         self.assertEqual(msg, failure.exception.__notes__[0])
 
     def test_custom_hex_error_is_noted(self):
         # Check hex codec gives a good error for malformed input
+        import binascii
         msg = "decoding with 'hex_codec' codec failed"
-        with self.assertRaises(Exception) as failure:
+        with self.assertRaises(binascii.Error) as failure:
             codecs.decode(b"hello", "hex_codec")
         self.assertEqual(msg, failure.exception.__notes__[0])
 
diff --git a/Lib/test/test_email/test_message.py b/Lib/test/test_email/test_message.py
index 4c754bf40fc..d3f396f02e7 100644
--- a/Lib/test/test_email/test_message.py
+++ b/Lib/test/test_email/test_message.py
@@ -696,14 +696,16 @@ def subtype_as_add(self, method, subtype, outcome):
             self.assertIsNone(part['Content-Disposition'])
 
     class _TestSetRaisingContentManager:
+        class CustomError(Exception):
+            pass
         def set_content(self, msg, content, *args, **kw):
-            raise Exception('test')
+            raise self.CustomError('test')
 
     def test_default_content_manager_for_add_comes_from_policy(self):
         cm = self._TestSetRaisingContentManager()
         m = self.message(policy=self.policy.clone(content_manager=cm))
         for method in ('add_related', 'add_alternative', 'add_attachment'):
-            with self.assertRaises(Exception) as ar:
+            with self.assertRaises(self._TestSetRaisingContentManager.CustomError) as ar:
                 getattr(m, method)('')
             self.assertEqual(str(ar.exception), 'test')
 
diff --git a/Lib/test/test_importlib/test_main.py b/Lib/test/test_importlib/test_main.py
index 6469aad7130..3b49227255e 100644
--- a/Lib/test/test_importlib/test_main.py
+++ b/Lib/test/test_importlib/test_main.py
@@ -68,7 +68,7 @@ def test_abc_enforced(self):
         dict(name=''),
     )
     def test_invalid_inputs_to_from_name(self, name):
-        with self.assertRaises(Exception):
+        with self.assertRaises(ValueError):
             Distribution.from_name(name)
 
 
diff --git a/Lib/test/test_mailbox.py b/Lib/test/test_mailbox.py
index 4c592eaf34d..4977a9369dd 100644
--- a/Lib/test/test_mailbox.py
+++ b/Lib/test/test_mailbox.py
@@ -116,10 +116,13 @@ def test_add_nonascii_string_header_raises(self):
         self.assertMailboxEmpty()
 
     def test_add_that_raises_leaves_mailbox_empty(self):
+        class CustomError(Exception): ...
+        exc_msg = "a fake error"
+
         def raiser(*args, **kw):
-            raise Exception("a fake error")
+            raise CustomError(exc_msg)
         support.patch(self, email.generator.BytesGenerator, 'flatten', raiser)
-        with self.assertRaises(Exception):
+        with self.assertRaisesRegex(CustomError, exc_msg):
             self._box.add(email.message_from_string("From: Alphöso"))
         self.assertEqual(len(self._box), 0)
         self._box.close()
diff --git a/Lib/test/test_shutil.py b/Lib/test/test_shutil.py
index 878a21a629f..93f20a6ff41 100644
--- a/Lib/test/test_shutil.py
+++ b/Lib/test/test_shutil.py
@@ -2738,7 +2738,7 @@ def test_regular_copy(self):
     def test_same_file(self):
         self.addCleanup(self.reset)
         with self.get_files() as (src, dst):
-            with self.assertRaises(Exception):
+            with self.assertRaises((OSError, _GiveupOnFastCopy)):
                 self.zerocopy_fun(src, src)
         # Make sure src file is not corrupted.
         self.assertEqual(read_file(TESTFN, binary=True), self.FILEDATA)
diff --git a/Lib/test/test_unittest/testmock/testasync.py b/Lib/test/test_unittest/testmock/testasync.py
index e9e1f63e84d..f57b83f457f 100644
--- a/Lib/test/test_unittest/testmock/testasync.py
+++ b/Lib/test/test_unittest/testmock/testasync.py
@@ -459,9 +459,10 @@ async def addition(self, var): pass
         self.assertEqual(output, 10)
 
     async def test_add_side_effect_exception(self):
+        class CustomError(Exception): pass
         async def addition(var): pass
-        mock = AsyncMock(addition, side_effect=Exception('err'))
-        with self.assertRaises(Exception):
+        mock = AsyncMock(addition, side_effect=CustomError('side-effect'))
+        with self.assertRaisesRegex(CustomError, 'side-effect'):
             await mock(5)
 
     async def test_add_side_effect_coroutine(self):