This paper analyzes the dynamical properties of a class of hybrid systems simulators. A hybrid system is a dynamical system with a state that can both flow and jump. Its simulator attempts to generate its solutions approximately. The paper presents mild regularity conditions on the hybrid system and its simulator to guarantee that simulated solutions are close to actual solutions on compact (hybrid) time intervals, and that asymptotically stable compact sets are preserved, in a semiglobal practical sense, under simulation. In fact, it is established that asymptotically stable compact sets are continuous in the integration step size parameter of the simulator; that is, as the step size of the Simulator converges to zero, the asymptotically stable set observed in simulations approaches the asymptotically stable compact set of the true hybrid system. Examples are used to illustrate concepts and results. (C) 2009 Elsevier Ltd. All rights reserved.