Hybrid dynamical systems, modeled by hybrid inclusions a combination of differential equations or inclusions, of difference equations or inclusions, and of constraints on the resulting motions are considered. Pointwise asymptotic stability, a property of a set of equilibria in a hybrid system where every equilibrium is Lyapunov stable and solutions from near the equilibria converge to some equilibrium, is studied. Sufficient conditions, relying on set-valued Lyapunov functions with strict or weak decrease, on invariance arguments, or on standard Lyapunov functions that also limit the lengths of solutions, are given. Structural properties of sets of solutions to a hybrid system, of reachable sets, and of limits of solutions are investigated in the presence of a pointwise asymptotically stable set of equilibria, and also under further uniform Zeno assumptions. Many of these results are extended to the case of partial pointwise asymptotic stability. The results are then used to extend Zeno solutions to hybrid systems beyond their Zeno times, in a way preserving reasonable dependence of solutions on initial conditions and enabling the analysis of convergence of extended solutions to a compact attractor.