It is the purpose of this talk to present recent advances in the numerical solution of piecewise smooth dynamics systems, described by systems of ordinary differential equations with discontinuous right-hand side. It is well known that such systems generate complex dynamics (e.g., crossing the discontinuity surfaces, sliding motion on surfaces of several co-dimensions, exiting from the discontinuity surfaces, etc.), which have to be accurately and efficiently handled by a reliable numerical scheme. With special emphasis on problems generating periodic orbits, and on sliding motion on the intersection of two discontinuity surfaces, we present a numerical approach based on an event-driven numerical scheme. The approach is based on a combination of a classical step-by-step strategy for the time integration with the detection of event points for the specification of the occurring dynamics. The main novelty of our approach is the systematic use of the ”moments regularization” to select a sliding vector field on the (intersection of) discontinuity surfaces. A dynamical study of the periodic behavior will also be provided, by means of the Floquet multipliers of the monodromy matrix. Numerical experiments will be presented, in order to confirm the effectiveness of the approach.