diff --git a/h2quic/server.go b/h2quic/server.go
index ec35179b0..254786eeb 100644
--- a/h2quic/server.go
+++ b/h2quic/server.go
@@ -13,6 +13,7 @@ import (
 
 	"github.com/lucas-clemente/quic-go"
 	"github.com/lucas-clemente/quic-go/protocol"
+	"github.com/lucas-clemente/quic-go/qerr"
 	"github.com/lucas-clemente/quic-go/utils"
 	"golang.org/x/net/http2"
 	"golang.org/x/net/http2/hpack"
@@ -104,7 +105,12 @@ func (s *Server) handleStream(session streamCreator, stream utils.Stream) {
 		var headerStreamMutex sync.Mutex // Protects concurrent calls to Write()
 		for {
 			if err := s.handleRequest(session, stream, &headerStreamMutex, hpackDecoder, h2framer); err != nil {
-				utils.Errorf("error handling h2 request: %s", err.Error())
+				// QuicErrors must originate from stream.Read() returning an error.
+				// In this case, the session has already logged the error, so we don't
+				// need to log it again.
+				if _, ok := err.(*qerr.QuicError); !ok {
+					utils.Errorf("error handling h2 request: %s", err.Error())
+				}
 				return
 			}
 		}
diff --git a/session.go b/session.go
index fe2e5e01e..c40b8651f 100644
--- a/session.go
+++ b/session.go
@@ -417,8 +417,16 @@ func (s *Session) closeImpl(e error, remoteClose bool) error {
 		e = qerr.PeerGoingAway
 	}
 
-	utils.Errorf("Closing session with error: %s", e.Error())
-	s.closeStreamsWithError(e)
+	quicErr := qerr.ToQuicError(e)
+
+	// Don't log 'normal' reasons
+	if quicErr.ErrorCode == qerr.PeerGoingAway || quicErr.ErrorCode == qerr.NetworkIdleTimeout {
+		utils.Infof("Closing connection %x", s.connectionID)
+	} else {
+		utils.Errorf("Closing session with error: %s", e.Error())
+	}
+
+	s.closeStreamsWithError(quicErr)
 	s.closeCallback(s.connectionID)
 
 	if remoteClose {
@@ -427,7 +435,6 @@ func (s *Session) closeImpl(e error, remoteClose bool) error {
 		return nil
 	}
 
-	quicErr := qerr.ToQuicError(e)
 	if quicErr.ErrorCode == qerr.DecryptionFailure {
 		// If we send a public reset, don't send a CONNECTION_CLOSE
 		s.closeChan <- nil
diff --git a/session_test.go b/session_test.go
index ed7af2112..42ba531d9 100644
--- a/session_test.go
+++ b/session_test.go
@@ -381,10 +381,10 @@ var _ = Describe("Session", func() {
 					Eventually(func() int { return runtime.NumGoroutine() }).Should(Equal(nGoRoutinesBefore))
 					n, err := s.Read([]byte{0})
 					Expect(n).To(BeZero())
-					Expect(err).To(MatchError(testErr))
+					Expect(err.Error()).To(ContainSubstring(testErr.Error()))
 					n, err = s.Write([]byte{0})
 					Expect(n).To(BeZero())
-					Expect(err).To(MatchError(testErr))
+					Expect(err.Error()).To(ContainSubstring(testErr.Error()))
 				})
 			})
 
@@ -603,7 +603,7 @@ var _ = Describe("Session", func() {
 				Expect(err).NotTo(HaveOccurred())
 				Eventually(func() bool { return atomic.LoadUint32(&session.closed) != 0 }).Should(BeTrue())
 				_, err = s.Write([]byte{})
-				Expect(err).To(MatchError(qerr.InvalidCryptoMessageType))
+				Expect(err.(*qerr.QuicError).ErrorCode).To(Equal(qerr.InvalidCryptoMessageType))
 			})
 
 			It("sends public reset after too many undecryptable packets", func() {