diff --git a/Lib/asyncio/base_events.py b/Lib/asyncio/base_events.py
index c58906f..9613ac2 100644
--- a/Lib/asyncio/base_events.py
+++ b/Lib/asyncio/base_events.py
@@ -1306,7 +1306,8 @@
                     if local_addr:
                         sock.bind(local_address)
                     if remote_addr:
-                        await self.sock_connect(sock, remote_address)
+                        if not allow_broadcast:
+                            await self.sock_connect(sock, remote_address)
                         r_addr = remote_address
                 except OSError as exc:
                     if sock is not None:
diff --git a/Lib/asyncio/selector_events.py b/Lib/asyncio/selector_events.py
index 93b6889..2996821 100644
--- a/Lib/asyncio/selector_events.py
+++ b/Lib/asyncio/selector_events.py
@@ -587,7 +587,10 @@
     def __init__(self, loop, sock, protocol, extra=None, server=None):
         super().__init__(extra, loop)
         self._extra['socket'] = sock
-        self._extra['sockname'] = sock.getsockname()
+        try:
+            self._extra['sockname'] = sock.getsockname()
+        except OSError:
+            self._extra['sockname'] = None
         if 'peername' not in self._extra:
             try:
                 self._extra['peername'] = sock.getpeername()
@@ -976,9 +979,11 @@
         if not data:
             return
 
-        if self._address and addr not in (None, self._address):
-            raise ValueError(
-                f'Invalid address: must be None or {self._address}')
+        if self._address:
+            if addr not in (None, self._address):
+                raise ValueError(
+                    f'Invalid address: must be None or {self._address}')
+            addr = self._address
 
         if self._conn_lost and self._address:
             if self._conn_lost >= constants.LOG_THRESHOLD_FOR_CONNLOST_WRITES:
@@ -989,7 +994,7 @@
         if not self._buffer:
             # Attempt to send it right away first.
             try:
-                if self._address:
+                if self._extra['peername']:
                     self._sock.send(data)
                 else:
                     self._sock.sendto(data, addr)
@@ -1012,7 +1017,7 @@
         while self._buffer:
             data, addr = self._buffer.popleft()
             try:
-                if self._address:
+                if self._extra['peername']:
                     self._sock.send(data)
                 else:
                     self._sock.sendto(data, addr)
diff --git a/Lib/test/test_asyncio/test_base_events.py b/Lib/test/test_asyncio/test_base_events.py
index c245c47..25420b2 100644
--- a/Lib/test/test_asyncio/test_base_events.py
+++ b/Lib/test/test_asyncio/test_base_events.py
@@ -1586,6 +1586,23 @@
         self.assertRaises(
             OSError, self.loop.run_until_complete, coro)
 
+    def test_create_datagram_endpoint_allow_broadcast(self):
+        protocol = MyDatagramProto(create_future=True, loop=self.loop)
+        self.loop.sock_connect = sock_connect = mock.Mock()
+        sock_connect.return_value = []
+
+        coro = self.loop.create_datagram_endpoint(
+            lambda: protocol,
+            remote_addr=('127.0.0.1', 0),
+            allow_broadcast=True)
+
+        transport, _ = self.loop.run_until_complete(coro)
+        self.assertFalse(sock_connect.called)
+
+        transport.close()
+        self.loop.run_until_complete(protocol.done)
+        self.assertEqual('CLOSED', protocol.state)
+
     @patch_socket
     def test_create_datagram_endpoint_socket_err(self, m_socket):
         m_socket.getaddrinfo = socket.getaddrinfo
diff --git a/Lib/test/test_asyncio/test_selector_events.py b/Lib/test/test_asyncio/test_selector_events.py
index d0d171a..bf721b0 100644
--- a/Lib/test/test_asyncio/test_selector_events.py
+++ b/Lib/test/test_asyncio/test_selector_events.py
@@ -1065,6 +1065,7 @@
         self.sock.fileno.return_value = 7
 
     def datagram_transport(self, address=None):
+        self.sock.getpeername.side_effect = None if address else OSError
         transport = _SelectorDatagramTransport(self.loop, self.sock,
                                                self.protocol,
                                                address=address)
diff --git a/Misc/NEWS.d/next/Library/2018-05-30-01-05-50.bpo-31922.fobsXJ.rst b/Misc/NEWS.d/next/Library/2018-05-30-01-05-50.bpo-31922.fobsXJ.rst
new file mode 100644
index 0000000..df3881b
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2018-05-30-01-05-50.bpo-31922.fobsXJ.rst
@@ -0,0 +1,3 @@
+:meth:`asyncio.AbstractEventLoop.create_datagram_endpoint`:
+Do not connect UDP socket when broadcast is allowed.
+This allows to receive replies after a UDP broadcast.