Modular networks of delay-coupled and pulse-coupled oscillators are presented, which display both transient (metastable) synchronization dynamics and the formation of a large number of "chimera" states characterized by coexistent synchronized and desynchronized subsystems. We consider networks based on both community and small-world topologies. It is shown through simulation that the metastable behaviour of the system is dependent in all cases on connection delay, and a critical region is found that maximizes indices of both metastability and the prevalence of chimera states. We show dependence of phase coherence in synchronous oscillation on the level and strength of external connectivity between communities, and demonstrate that synchronization dynamics are dependent on the modular structure of the network. The long-term behaviour of the system is considered and the relevance of the model briefly discussed with emphasis on biological and neurobiological systems.