Systems that have the structure of a network of interconnected nodes are abundant in nature and technology. Many mathematical models of such networks are given as ordinary differential equations defined by smooth vector fields. These traditional or "synchronous" network models, however, fail to incorporate nonsmooth features seen in real world; for example, individual nodes of a network cannot stop and restart in finite time. Asynchronous networks are an attempt to set up a mathematical framework to study systems that exhibit stopping of nodes and their bifurcations. We also consider asynchronous networks with function: given some set of initial conditions a desired final output state has to be reached. We discuss deadlocks -- dnamical states that prevent a network to complete its function -- and indicate how functional networks can be decomposed into simpler components. Joint work with Mike Field.