diff --git a/irctest/server_tests/messages.py b/irctest/server_tests/messages.py
index b33be17..c815e94 100644
--- a/irctest/server_tests/messages.py
+++ b/irctest/server_tests/messages.py
@@ -32,6 +32,26 @@ class PrivmsgTestCase(cases.BaseServerTestCase):
         # ERR_NOSUCHNICK, ERR_NOSUCHCHANNEL, or ERR_CANNOTSENDTOCHAN
         self.assertIn(msg.command, ("401", "403", "404"))
 
+    @cases.mark_specifications("RFC1459", "RFC2812")
+    def testPrivmsgToUser(self):
+        """<https://tools.ietf.org/html/rfc2812#section-3.3.1>"""
+        self.connectClient("foo")
+        self.connectClient("bar")
+        self.sendLine(1, "PRIVMSG bar :hey there!")
+        self.getMessages(1)
+        pms = [msg for msg in self.getMessages(2) if msg.command == "PRIVMSG"]
+        self.assertEqual(len(pms), 1)
+        self.assertMessageMatch(pms[0], command="PRIVMSG", params=["bar", "hey there!"])
+
+    @cases.mark_specifications("RFC1459", "RFC2812")
+    def testPrivmsgNonexistentUser(self):
+        """https://tools.ietf.org/html/rfc2812#section-3.3.1"""
+        self.connectClient("foo")
+        self.sendLine(1, "PRIVMSG bar :hey there!")
+        msg = self.getMessage(1)
+        # ERR_NOSUCHNICK
+        self.assertIn(msg.command, ("401"))
+
 
 class NoticeTestCase(cases.BaseServerTestCase):
     @cases.mark_specifications("RFC1459", "RFC2812")