diff --git a/Lib/test/support/__init__.py b/Lib/test/support/__init__.py
index 19ea9764e96..a65de4a5abe 100644
--- a/Lib/test/support/__init__.py
+++ b/Lib/test/support/__init__.py
@@ -1493,6 +1493,8 @@ def get_socket_conn_refused_errs():
         # bpo-31910: socket.create_connection() fails randomly
         # with EADDRNOTAVAIL on Travis CI
         errors.append(errno.EADDRNOTAVAIL)
+    if not IPV6_ENABLED:
+        errors.append(errno.EAFNOSUPPORT)
     return errors
 
 
diff --git a/Lib/test/test_asyncio/test_base_events.py b/Lib/test/test_asyncio/test_base_events.py
index 811b37425dd..08d4792fa72 100644
--- a/Lib/test/test_asyncio/test_base_events.py
+++ b/Lib/test/test_asyncio/test_base_events.py
@@ -91,6 +91,9 @@ class BaseEventTests(test_utils.TestCase):
         self.assertIsNone(
             base_events._ipaddr_info('1.2.3.4', 1, UNSPEC, 0, 0))
 
+        if not support.IPV6_ENABLED:
+            return
+
         # IPv4 address with family IPv6.
         self.assertIsNone(
             base_events._ipaddr_info('1.2.3.4', 1, INET6, STREAM, TCP))
@@ -1149,7 +1152,7 @@ class BaseEventLoopWithSelectorTests(test_utils.TestCase):
             srv.close()
             self.loop.run_until_complete(srv.wait_closed())
 
-    @unittest.skipUnless(hasattr(socket, 'AF_INET6'), 'no IPv6 support')
+    @unittest.skipUnless(support.IPV6_ENABLED, 'no IPv6 support')
     def test_create_server_ipv6(self):
         async def main():
             with self.assertWarns(DeprecationWarning):
@@ -1281,6 +1284,9 @@ class BaseEventLoopWithSelectorTests(test_utils.TestCase):
             t.close()
             test_utils.run_briefly(self.loop)  # allow transport to close
 
+        if not support.IPV6_ENABLED:
+            return
+
         sock.family = socket.AF_INET6
         coro = self.loop.create_connection(asyncio.Protocol, '::1', 80)
         t, p = self.loop.run_until_complete(coro)
@@ -1298,6 +1304,7 @@ class BaseEventLoopWithSelectorTests(test_utils.TestCase):
             t.close()
             test_utils.run_briefly(self.loop)  # allow transport to close
 
+    @unittest.skipUnless(support.IPV6_ENABLED, 'no IPv6 support')
     @unittest.skipIf(sys.platform.startswith('aix'),
                     "bpo-25545: IPv6 scope id and getaddrinfo() behave differently on AIX")
     @patch_socket
diff --git a/Lib/test/test_socket.py b/Lib/test/test_socket.py
index 50094de58bf..e92f871880a 100644
--- a/Lib/test/test_socket.py
+++ b/Lib/test/test_socket.py
@@ -4964,8 +4964,15 @@ class NetworkConnectionNoServer(unittest.TestCase):
         # Issue #9792: create_connection() should not recast timeout errors
         # as generic socket errors.
         with self.mocked_socket_module():
-            with self.assertRaises(socket.timeout):
+            try:
                 socket.create_connection((HOST, 1234))
+            except socket.timeout:
+                pass
+            except OSError as exc:
+                if support.IPV6_ENABLED or exc.errno != errno.EAFNOSUPPORT:
+                    raise
+            else:
+                self.fail('socket.timeout not raised')
 
 
 class NetworkConnectionAttributesTest(SocketTCPTest, ThreadableTest):
diff --git a/Lib/test/test_ssl.py b/Lib/test/test_ssl.py
index c72a85710d5..064f0e8d4de 100644
--- a/Lib/test/test_ssl.py
+++ b/Lib/test/test_ssl.py
@@ -676,7 +676,7 @@ class BasicSocketTests(unittest.TestCase):
         fail(cert, 'example.net')
 
         # -- IPv6 matching --
-        if hasattr(socket, 'AF_INET6'):
+        if support.IPV6_ENABLED:
             cert = {'subject': ((('commonName', 'example.com'),),),
                     'subjectAltName': (
                         ('DNS', 'example.com'),
@@ -757,7 +757,7 @@ class BasicSocketTests(unittest.TestCase):
                 ssl._inet_paton(invalid)
         for ipaddr in ['127.0.0.1', '192.168.0.1']:
             self.assertTrue(ssl._inet_paton(ipaddr))
-        if hasattr(socket, 'AF_INET6'):
+        if support.IPV6_ENABLED:
             for ipaddr in ['::1', '2001:db8:85a3::8a2e:370:7334']:
                 self.assertTrue(ssl._inet_paton(ipaddr))
 
diff --git a/Misc/NEWS.d/next/Tests/2019-06-29-23-56-28.bpo-37199.FHDsLf.rst b/Misc/NEWS.d/next/Tests/2019-06-29-23-56-28.bpo-37199.FHDsLf.rst
new file mode 100644
index 00000000000..b05209159cc
--- /dev/null
+++ b/Misc/NEWS.d/next/Tests/2019-06-29-23-56-28.bpo-37199.FHDsLf.rst
@@ -0,0 +1 @@
+Fix test failures when IPv6 is unavailable or disabled.