The temporal communication graph is a new graph theoretic model of parallel computation that we have developed for the mapping of parallel computations to message-passing parallel architectures. The TCG can be viewed as an augmented version of Lamport's process-time graphs. It integrates the two dominant models currently in use in the areas of mapping and scheduling: the static task graph and the DAG. In addition, the TCG provides the capability to identify logically synchronous phases of communication and computation, and to describe the temporal behavior of a parallel algorithm in terms of these phases. This paper defines the TCG, introduces a language for describing TCGs, illustrates its use for mapping and scheduling, and discusses a wide range of potential uses for the TCG in the area of parallel programming environments.