  In this paper we define and study the billiard problem on bounded regions on
surfaces of constant curvature. We show that this problem defines a
2-dimensional conservative and reversible dynamical system, defined by a Twist
diffeomorphism, if the boundary of the region is an oval. Using these
properties and defining good perturbations for billiards, we show, in this new
version, that having only a finite number of nondegenerate periodic orbits for
each fixed period is an open property for billiards on surfaces of constant
curvature and a dense one on the Euclidean and the hyperbolic planes. For the
proof of the density, the techniques we use for the Euclidean and hyperbolic
cases, do not work for the spherical case, due to a constraint (the perimeter
of the polygonal trajectory being a multiple of {\pi}). We finish this paper
studying the stability of these nondegenerate orbits.
