diff --git a/server.go b/server.go
index 40804ea..1431251 100644
--- a/server.go
+++ b/server.go
@@ -452,6 +452,11 @@ func (c *serverConn) run(sctx context.Context) {
 			if err != nil && err != io.EOF {
 				logrus.WithError(err).Error("error receiving message")
 			}
+			if err == io.EOF || err == io.ErrUnexpectedEOF {
+				// The client went away and we should stop processing
+				// requests, so that the client connection is closed
+				return
+			}
 		case <-shutdown:
 			return
 		}
diff --git a/server_test.go b/server_test.go
index 4f31feb..9ee7e0d 100644
--- a/server_test.go
+++ b/server_test.go
@@ -28,6 +28,7 @@ import (
 
 	"github.com/gogo/protobuf/proto"
 	"github.com/pkg/errors"
+	"github.com/prometheus/procfs"
 	"google.golang.org/grpc/codes"
 	"google.golang.org/grpc/status"
 )
@@ -359,6 +360,48 @@ func TestClientEOF(t *testing.T) {
 	}
 }
 
+func TestServerEOF(t *testing.T) {
+	var (
+		ctx             = context.Background()
+		server          = mustServer(t)(NewServer())
+		addr, listener  = newTestListener(t)
+		client, cleanup = newTestClient(t, addr)
+	)
+	defer cleanup()
+	defer listener.Close()
+
+	socketCountBefore := socketCount(t)
+
+	go server.Serve(ctx, listener)
+
+	registerTestingService(server, &testingServer{})
+
+	tp := &testPayload{}
+	// do a regular call
+	if err := client.Call(ctx, serviceName, "Test", tp, tp); err != nil {
+		t.Fatalf("unexpected error during test call: %v", err)
+	}
+
+	// close the client, so that server gets EOF
+	if err := client.Close(); err != nil {
+		t.Fatalf("unexpected error while closing client: %v", err)
+	}
+
+	// server should eventually close the client connection
+	maxAttempts := 20
+	for i := 1; i <= maxAttempts; i++ {
+		socketCountAfter := socketCount(t)
+		if socketCountAfter < socketCountBefore {
+			break
+		}
+		if i == maxAttempts {
+			t.Fatalf("expected number of open sockets to be less than %d after client close, got %d open sockets",
+				socketCountBefore, socketCountAfter)
+		}
+		time.Sleep(100 * time.Millisecond)
+	}
+}
+
 func TestUnixSocketHandshake(t *testing.T) {
 	var (
 		ctx             = context.Background()
@@ -541,3 +584,22 @@ func mustServer(t testing.TB) func(server *Server, err error) *Server {
 		return server
 	}
 }
+
+func socketCount(t *testing.T) int {
+	proc, err := procfs.Self()
+	if err != nil {
+		t.Fatalf("unexpected error while reading procfs: %v", err)
+	}
+	fds, err := proc.FileDescriptorTargets()
+	if err != nil {
+		t.Fatalf("unexpected error while listing open file descriptors: %v", err)
+	}
+
+	sockets := 0
+	for _, fd := range fds {
+		if strings.Contains(fd, "socket") {
+			sockets++
+		}
+	}
+	return sockets
+}