We present Lyapunov stability results for discontinuous dynamical systems (DDS) determined by linear and nonlinear semigroups defined on Banach space. DDS of the type considered herein arise in the modeling of a variety of finite- and infinite-dimensional systems, including certain classes of hybrid systems, discrete-event systems, switched systems, systems subjected to impulse effects, and the like. We apply our results in the analysis of several important specific classes of DDS.