Up to recently the majority of applications of the Abstract State Machines method for design and verification of computational systems used the shared variable approach. However in particular with distributed systems only various forms of communication may be available to share information. We define communicating ASMs by using instead of shared locations an explicit, abstract concept of Sending/Receiving messages which can be added to existing ASM execution engines. We aim to provide a definition which is a conservative extension of traditional ASMs, uniformly captures synchronous and asynchronous communication and is not bound to any specific message passing mechanism but can be instantiated to the major communication concepts in the literature. We illustrate the concept by an application to synchronous and asynchronous networks of communicating processes.