In this paper, we consider the synchronization problem for a network of state diffusive coupled agents with identical but time-varying dynamics under switching communication topologies. Both leader-follower and leaderless networks are studied. For leader-follower networks, sufficient conditions are established regarding the connectivity of communication graphs for the network to achieve global exponential or asymptotic synchronization, respectively, for directed and undirected communications. For leaderless networks, sufficient conditions for asymptotic norm synchronization are established, respectively, for directed and undirected communications, in the sense that the norm of agents' states converges to a common value. We also show that if the communication graph is fixed, norm synchronization leads to state synchronization. The results reveal some important similarities and differences between leaderless and leader-follower networks for synchronization under directed switching communication graphs.