The dynamics associated with mechanical Hamiltonian flows with smooth potentials that include sharp fronts may be modeled, at the singular limit, by Hamiltonian impact systems: a class of generalized billiards by which the dynamics in the domain’s interior are governed by smooth potentials and at the domain’s boundaries by elastic reflections. Results on persisting vs. non-persisting dynamics of such systems will be discussed. In some cases, called quasi-integrable, the limit systems have fascinating behavior: their energy surfaces are foliated by two dimensional level sets. The motion on each of these level sets is conjugated to a directed motion on a translation surface. The genus of the iso-energy level sets varies - it is only piecewise constant along the foliation. The metric data of the corresponding translation surfaces and the direction of motion along them changes smoothly within each of the constant-genus families. Ergodic properties and quantum properties of classes of such systems are established. Based on collaborations with: M. Pnueli, K. Fraczek, O. Yaniv, D. Turaev, L. Becker, S. Elliott, B. Firester, S. Gonen Cohen.