diff --git a/tornado/test/iostream_test.py b/tornado/test/iostream_test.py
index e22e83e60f..02fcd3e13f 100644
--- a/tornado/test/iostream_test.py
+++ b/tornado/test/iostream_test.py
@@ -1149,6 +1149,88 @@ def handle_stream(self, stream, address):
         yield handshake_future
 
 
+class TestIOStreamCheckHostname(AsyncTestCase):
+    # This test ensures that hostname checks are working correctly after
+    # #3337 revealed that we have no test coverage in this area, and we
+    # removed a manual hostname check that was needed only for very old
+    # versions of python.
+    def setUp(self):
+        super().setUp()
+        self.listener, self.port = bind_unused_port()
+
+        def accept_callback(connection, address):
+            ssl_ctx = ssl.create_default_context(ssl.Purpose.CLIENT_AUTH)
+            ssl_ctx.load_cert_chain(
+                os.path.join(os.path.dirname(__file__), "test.crt"),
+                os.path.join(os.path.dirname(__file__), "test.key"),
+            )
+            connection = ssl_ctx.wrap_socket(
+                connection,
+                server_side=True,
+                do_handshake_on_connect=False,
+            )
+            SSLIOStream(connection)
+
+        netutil.add_accept_handler(self.listener, accept_callback)
+
+        # Our self-signed cert is its own CA.  We have to pass the CA check before
+        # the hostname check will be performed.
+        self.client_ssl_ctx = ssl.create_default_context(ssl.Purpose.SERVER_AUTH)
+        self.client_ssl_ctx.load_verify_locations(
+            os.path.join(os.path.dirname(__file__), "test.crt")
+        )
+
+    def tearDown(self):
+        self.io_loop.remove_handler(self.listener.fileno())
+        self.listener.close()
+        super().tearDown()
+
+    @gen_test
+    async def test_match(self):
+        stream = SSLIOStream(socket.socket(), ssl_options=self.client_ssl_ctx)
+        await stream.connect(
+            ("127.0.0.1", self.port),
+            server_hostname="foo.example.com",
+        )
+        stream.close()
+
+    @gen_test
+    async def test_no_match(self):
+        stream = SSLIOStream(socket.socket(), ssl_options=self.client_ssl_ctx)
+        with ExpectLog(
+            gen_log,
+            ".*alert bad certificate",
+            level=logging.WARNING,
+            required=platform.system() != "Windows",
+        ):
+            with self.assertRaises(ssl.SSLCertVerificationError):
+                with ExpectLog(
+                    gen_log,
+                    ".*(certificate verify failed: Hostname mismatch)",
+                    level=logging.WARNING,
+                ):
+                    await stream.connect(
+                        ("127.0.0.1", self.port),
+                        server_hostname="bar.example.com",
+                    )
+            # The server logs a warning while cleaning up the failed connection.
+            # Unfortunately there's no good hook to wait for this logging.
+            # It doesn't seem to happen on windows; I'm not sure why.
+            if platform.system() != "Windows":
+                await asyncio.sleep(0.1)
+
+    @gen_test
+    async def test_check_disabled(self):
+        # check_hostname can be set to false and the connection will succeed even though it doesn't
+        # have the right hostname.
+        self.client_ssl_ctx.check_hostname = False
+        stream = SSLIOStream(socket.socket(), ssl_options=self.client_ssl_ctx)
+        await stream.connect(
+            ("127.0.0.1", self.port),
+            server_hostname="bar.example.com",
+        )
+
+
 @skipIfNonUnix
 class TestPipeIOStream(TestReadWriteMixin, AsyncTestCase):
     @gen.coroutine
diff --git a/tornado/test/test.crt b/tornado/test/test.crt
index ffc49b06c1..c7f19e3e62 100644
--- a/tornado/test/test.crt
+++ b/tornado/test/test.crt
@@ -1,20 +1,18 @@
 -----BEGIN CERTIFICATE-----
-MIIDWzCCAkOgAwIBAgIUV4spou0CenmvKqa7Hml/MC+JKiAwDQYJKoZIhvcNAQEL
-BQAwPTELMAkGA1UEBhMCVVMxEzARBgNVBAgMCkNhbGlmb3JuaWExGTAXBgNVBAoM
-EFRvcm5hZG8gV2ViIFRlc3QwHhcNMTgwOTI5MTM1NjQ1WhcNMjgwOTI2MTM1NjQ1
-WjA9MQswCQYDVQQGEwJVUzETMBEGA1UECAwKQ2FsaWZvcm5pYTEZMBcGA1UECgwQ
-VG9ybmFkbyBXZWIgVGVzdDCCASIwDQYJKoZIhvcNAQEBBQADggEPADCCAQoCggEB
-AKT0LdyI8tW5uwP3ahE8BFSz+j3SsKBDv/0cKvqxVVE6sLEST2s3HjArZvIIG5sb
-iBkWDrqnZ6UKDvB4jlobLGAkepxDbrxHWxK53n0C28XXGLqJQ01TlTZ5rpjttMeg
-5SKNjHbxpOvpUwwQS4br4WjZKKyTGiXpFkFUty+tYVU35/U2yyvreWHmzpHx/25t
-H7O2RBARVwJYKOGPtlH62lQjpIWfVfklY4Ip8Hjl3B6rBxPyBULmVQw0qgoZn648
-oa4oLjs0wnYBz01gVjNMDHej52SsB/ieH7W1TxFMzqOlcvHh41uFbQJPgcXsruSS
-9Z4twzSWkUp2vk/C//4Sz38CAwEAAaNTMFEwHQYDVR0OBBYEFLf8fQ5+u8sDWAd3
-r5ZjZ5MmDWJeMB8GA1UdIwQYMBaAFLf8fQ5+u8sDWAd3r5ZjZ5MmDWJeMA8GA1Ud
-EwEB/wQFMAMBAf8wDQYJKoZIhvcNAQELBQADggEBADkkm3pIb9IeqVNmQ2uhQOgw
-UwyToTYUHNTb/Nm5lzBTBqC8gbXAS24RQ30AB/7G115Uxeo+YMKfITxm/CgR+vhF
-F59/YrzwXj+G8bdbuVl/UbB6f9RSp+Zo93rUZAtPWr77gxLUrcwSRzzDwxFjC2nC
-6eigbkvt1OQY775RwnFAt7HKPclE0Out+cGJIboJuO1f3r57ZdyFH0GzbZEff/7K
-atGXohijWJjYvU4mk0KFHORZrcBpsv9cfkFbmgVmiRwxRJ1tLauHM3Ne+VfqYE5M
-4rTStSyz3ASqVKJ2iFMQueNR/tUOuDlfRt+0nhJMuYSSkW+KTgnwyOGU9cv+mxA=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 -----END CERTIFICATE-----
diff --git a/tornado/test/test.key b/tornado/test/test.key
index 7cb7d8d23c..8eea05db5f 100644
--- a/tornado/test/test.key
+++ b/tornado/test/test.key
@@ -1,28 +1,28 @@
 -----BEGIN PRIVATE KEY-----
-MIIEvQIBADANBgkqhkiG9w0BAQEFAASCBKcwggSjAgEAAoIBAQCk9C3ciPLVubsD
-92oRPARUs/o90rCgQ7/9HCr6sVVROrCxEk9rNx4wK2byCBubG4gZFg66p2elCg7w
-eI5aGyxgJHqcQ268R1sSud59AtvF1xi6iUNNU5U2ea6Y7bTHoOUijYx28aTr6VMM
-EEuG6+Fo2Siskxol6RZBVLcvrWFVN+f1Nssr63lh5s6R8f9ubR+ztkQQEVcCWCjh
-j7ZR+tpUI6SFn1X5JWOCKfB45dweqwcT8gVC5lUMNKoKGZ+uPKGuKC47NMJ2Ac9N
-YFYzTAx3o+dkrAf4nh+1tU8RTM6jpXLx4eNbhW0CT4HF7K7kkvWeLcM0lpFKdr5P
-wv/+Es9/AgMBAAECggEABi6AaXtYXloPgB6NgwfUwbfc8OQsalUfpMShd7OdluW0
-KW6eO05de0ClIvzay/1EJGyHMMeFQtIVrT1XWFkcWJ4FWkXMqJGkABenFtg8lDVz
-X8o1E3jGZrw4ptKBq9mDvL/BO9PiclTUH+ecbPn6AIvi0lTQ7grGIryiAM9mjmLy
-jpCwoutF2LD4RPNg8vqWe/Z1rQw5lp8FOHhRwPooHHeoq1bSrp8dqvVAwAam7Mmf
-uFgI8jrNycPgr2cwEEtbq2TQ625MhVnCpwT+kErmAStfbXXuqv1X1ZZgiNxf+61C
-OL0bhPRVIHmmjiK/5qHRuN4Q5u9/Yp2SJ4W5xadSQQKBgQDR7dnOlYYQiaoPJeD/
-7jcLVJbWwbr7bE19O/QpYAtkA/FtGlKr+hQxPhK6OYp+in8eHf+ga/NSAjCWRBoh
-MNAVCJtiirHo2tFsLFOmlJpGL9n3sX8UnkJN90oHfWrzJ8BZnXaSw2eOuyw8LLj+
-Q+ISl6Go8/xfsuy3EDv4AP1wCwKBgQDJJ4vEV3Kr+bc6N/xeu+G0oHvRAWwuQpcx
-9D+XpnqbJbFDnWKNE7oGsDCs8Qjr0CdFUN1pm1ppITDZ5N1cWuDg/47ZAXqEK6D1
-z13S7O0oQPlnsPL7mHs2Vl73muAaBPAojFvceHHfccr7Z94BXqKsiyfaWz6kclT/
-Nl4JTdsC3QKBgQCeYgozL2J/da2lUhnIXcyPstk+29kbueFYu/QBh2HwqnzqqLJ4
-5+t2H3P3plQUFp/DdDSZrvhcBiTsKiNgqThEtkKtfSCvIvBf4a2W/4TJsW6MzxCm
-2KQDuK/UqM4Y+APKWN/N6Lln2VWNbNyBkWuuRVKFatccyJyJnSjxeqW7cwKBgGyN
-idCYPIrwROAHLItXKvOWE5t0ABRq3TsZC2RkdA/b5HCPs4pclexcEriRjvXrK/Yt
-MH94Ve8b+UftSUQ4ytjBMS6MrLg87y0YDhLwxv8NKUq65DXAUOW+8JsAmmWQOqY3
-MK+m1BT4TMklgVoN3w3sPsKIsSJ/jLz5cv/kYweFAoGAG4iWU1378tI2Ts/Fngsv
-7eoWhoda77Y9D0Yoy20aN9VdMHzIYCBOubtRPEuwgaReNwbUBWap01J63yY/fF3K
-8PTz6covjoOJqxQJOvM7nM0CsJawG9ccw3YXyd9KgRIdSt6ooEhb7N8W2EXYoKl3
-g1i2t41Q/SC3HUGC5mJjpO8=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 -----END PRIVATE KEY-----
diff --git a/tornado/testing.py b/tornado/testing.py
index da33335733..bdbff87bc3 100644
--- a/tornado/testing.py
+++ b/tornado/testing.py
@@ -517,7 +517,9 @@ def get_ssl_options(self) -> Dict[str, Any]:
     def default_ssl_options() -> Dict[str, Any]:
         # Testing keys were generated with:
         # openssl req -new -keyout tornado/test/test.key \
-        #                     -out tornado/test/test.crt -nodes -days 3650 -x509
+        #     -out tornado/test/test.crt \
+        #     -nodes -days 3650 -x509 \
+        #     -subj "/CN=foo.example.com" -addext "subjectAltName = DNS:foo.example.com"
         module_dir = os.path.dirname(__file__)
         return dict(
             certfile=os.path.join(module_dir, "test", "test.crt"),