diff --git a/tests/test_udp.py b/tests/test_udp.py
index af330462..d9d6e018 100644
--- a/tests/test_udp.py
+++ b/tests/test_udp.py
@@ -56,7 +56,8 @@ def datagram_received(self, data, addr):
 
         s_transport, server = self.loop.run_until_complete(coro)
 
-        host, port, *_ = s_transport.get_extra_info('sockname')
+        remote_addr = s_transport.get_extra_info('sockname')
+        host, port, *_ = remote_addr
 
         self.assertIsInstance(server, TestMyDatagramProto)
         self.assertEqual('INITIALIZED', server.state)
@@ -86,6 +87,36 @@ def datagram_received(self, data, addr):
         # received
         self.assertEqual(8, client.nbytes)
 
+        # https://github.com/MagicStack/uvloop/issues/319
+        # uvloop should behave the same as asyncio when given remote_addr
+        transport.sendto(b'xxx', remote_addr)
+        tb.run_until(
+            self.loop, lambda: server.nbytes > 3 or client.done.done())
+        self.assertEqual(6, server.nbytes)
+        tb.run_until(self.loop, lambda: client.nbytes > 8)
+
+        # received
+        self.assertEqual(16, client.nbytes)
+
+        # reject sendto with a different port
+        with self.assertRaisesRegex(
+            ValueError, "Invalid address.*" + repr(remote_addr)
+        ):
+            bad_addr = list(remote_addr)
+            bad_addr[1] += 1
+            bad_addr = tuple(bad_addr)
+            transport.sendto(b"xxx", bad_addr)
+
+        # reject sento with unresolved hostname
+        if remote_addr[0] != lc_addr[0]:
+            with self.assertRaisesRegex(
+                ValueError, "Invalid address.*" + repr(remote_addr)
+            ):
+                bad_addr = list(remote_addr)
+                bad_addr[0] = lc_addr[0]
+                bad_addr = tuple(bad_addr)
+                transport.sendto(b"xxx", bad_addr)
+
         # extra info is available
         self.assertIsNotNone(transport.get_extra_info('sockname'))
 
@@ -100,8 +131,11 @@ def test_create_datagram_endpoint_addrs_ipv4(self):
         self._test_create_datagram_endpoint_addrs(
             socket.AF_INET, ('127.0.0.1', 0))
 
-    @unittest.skipUnless(tb.has_IPv6, 'no IPv6')
-    def test_create_datagram_endpoint_addrs_ipv6(self):
+    def test_create_datagram_endpoint_addrs_ipv4_nameaddr(self):
+        self._test_create_datagram_endpoint_addrs(
+            socket.AF_INET, ('localhost', 0))
+
+    def _test_create_datagram_endpoint_addrs_ipv6(self):
         self._test_create_datagram_endpoint_addrs(
             socket.AF_INET6, ('::1', 0))
 
@@ -291,6 +325,16 @@ def test_send_after_close(self):
         self.loop.run_until_complete(asyncio.sleep(0.01))
         s_transport.sendto(b'aaaa', ('127.0.0.1', 80))
 
+    @unittest.skipUnless(tb.has_IPv6, 'no IPv6')
+    def test_create_datagram_endpoint_addrs_ipv6(self):
+        self._test_create_datagram_endpoint_addrs_ipv6()
+
 
 class Test_AIO_UDP(_TestUDP, tb.AIOTestCase):
-    pass
+    @unittest.skipUnless(tb.has_IPv6, 'no IPv6')
+    @unittest.skipIf(
+        sys.version_info[:3] < (3, 6, 7) or sys.version_info[:3] == (3, 7, 0),
+        'bpo-27500: bug fixed in Python 3.6.7, 3.7.1 and above.',
+    )
+    def test_create_datagram_endpoint_addrs_ipv6(self):
+        self._test_create_datagram_endpoint_addrs_ipv6()
diff --git a/uvloop/handles/udp.pxd b/uvloop/handles/udp.pxd
index 0ef68b22..0647f23e 100644
--- a/uvloop/handles/udp.pxd
+++ b/uvloop/handles/udp.pxd
@@ -2,8 +2,10 @@ cdef class UDPTransport(UVBaseTransport):
     cdef:
         bint __receiving
         int _family
+        object _address
 
     cdef _init(self, Loop loop, unsigned int family)
+    cdef _set_address(self, system.addrinfo *addr)
 
     cdef _connect(self, system.sockaddr* addr, size_t addr_len)
 
diff --git a/uvloop/handles/udp.pyx b/uvloop/handles/udp.pyx
index 675407d7..3bca6205 100644
--- a/uvloop/handles/udp.pyx
+++ b/uvloop/handles/udp.pyx
@@ -55,6 +55,7 @@ cdef class UDPTransport(UVBaseTransport):
     def __cinit__(self):
         self._family = uv.AF_UNSPEC
         self.__receiving = 0
+        self._address = None
 
     cdef _init(self, Loop loop, unsigned int family):
         cdef int err
@@ -78,6 +79,9 @@ cdef class UDPTransport(UVBaseTransport):
 
         self._finish_init()
 
+    cdef _set_address(self, system.addrinfo *addr):
+        self._address = __convert_sockaddr_to_pyaddr(addr.ai_addr)
+
     cdef _connect(self, system.sockaddr* addr, size_t addr_len):
         cdef int err
         err = uv.uv_udp_connect(<uv.uv_udp_t*>self._handle, addr)
@@ -279,6 +283,16 @@ cdef class UDPTransport(UVBaseTransport):
             # Replicating asyncio logic here.
             return
 
+        if self._address:
+            if addr not in (None, self._address):
+                # Replicating asyncio logic here.
+                raise ValueError(
+                    'Invalid address: must be None or %s' % (self._address,))
+
+            # Instead of setting addr to self._address below like what asyncio
+            # does, we depend on previous uv_udp_connect() to set the address
+            addr = None
+
         if self._conn_lost:
             # Replicating asyncio logic here.
             if self._conn_lost >= LOG_THRESHOLD_FOR_CONNLOST_WRITES:
diff --git a/uvloop/loop.pyx b/uvloop/loop.pyx
index a368c523..c9bb650d 100644
--- a/uvloop/loop.pyx
+++ b/uvloop/loop.pyx
@@ -3004,6 +3004,7 @@ cdef class Loop:
                     rai = (<AddrInfo>rads).data
                     udp._init(self, rai.ai_family)
                     udp._connect(rai.ai_addr, rai.ai_addrlen)
+                    udp._set_address(rai)
                 else:
                     if family not in (uv.AF_INET, uv.AF_INET6):
                         raise ValueError('unexpected address family')
@@ -3047,6 +3048,7 @@ cdef class Loop:
                             rai = rai.ai_next
                             continue
                         udp._connect(rai.ai_addr, rai.ai_addrlen)
+                        udp._set_address(rai)
                         break
                     else:
                         raise OSError(