A theoretical framework for the nonequilibrium statistical mechanics of open systems is constructed. This is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary system S in interaction with a ``large'' reservoir R. The dynamics of S are analyzed on the basis of a precise quantum-mechanical treatment of the microscopic equations of motion for the combined system S + R. On proceeding to the thermodynamical limit for R we obtain a generalized master equation for S, subject to specified conditions on the many-particle structure of R, its initial state, and its coupling to S. This master equation corresponds to a self-contained law of motion for S, in which the R variables appear only in the forms of certain thermal averages, taken over the initial state. This dynamical law is a generalization of the quantum-mechanical Liouville equation to a form appropriate to open systems.