Our talk is concerned with the study of dynamical systems which are piecewise contraction maps (PC maps). Certain mathematical models, like contracting outer billi- ards and switched flow systems, are described by PC maps. In the setup, one considers a convex subset X ⊂ Rd and a finite partition of X: X1,...,Xn. Then the map f : X → X is assumed to be a contraction on each element Xi of the partition. It is expected that a typical PC map has finitely many periodic orbits and every orbit converges to a periodic orbit. We will show that, under certain conditions, a typical PC map, in the metrical sense of the parameter set, is indeed asymptotically periodic, in particular, this was established for one-dimensional PC maps in a joint work with Benito Pires and Rafael Rosales.