Hybrid systems are richly expressive models for a large variety of potential ap- plications. However, being so rich as to include continuous nonlinear dynamical systems, discrete-event systems and other models of computation (unite-state machines and data ow come to mind here), they are not amenable to com- putationally attractive techniques for synthesis and analysis and present hard numerical problems to simulation. Hence, applying the methods typical of this technology requires non trivial amount of approximation and abstraction. And approximation and abstraction are effective only if the domain of application is deeply understood. Thus, significant applications of hybrid systems require a great deal of work both to select the right abstraction level and to derive algo- rithms that exploit the particularities of the domain of application. In addition, one needs to motivate and document convincingly why using hybrid systems can yield better results than other techniques. In this respect, there has been an on- going debate as to what constitutes a meaningful result in applications: on one hand, novel languages for describing hybrid systems and capturing their prop- erties may be considered sophomoric exercises by experts in languages, on the other, formal verification tools that in general can handle small systems may be seen as toys for who is trying to tame entire chemical plants. On the simulation front, how to deal with discontinuities of trajectories is a major issue. Numerical analysts have been looking at these problems only recently and with a great deal of skepticism as to what can be proven rigorously. Hybrid system researchers are now getting seriously in the simulation arena exploiting what has been done in the numerical analysis arena.