Let P be a simple, closed polygon in the plane, all interior angles of which are rational multiples of π. We consider the possible paths of a point, rebounding in the interior of P with constant speed and elastic reflections. Such a dynamical system is known as "billiards in P". By means of a well-known construction, "billiard" trajectories in such a polygon P are identified with geodesic paths on a closed Riemann surface X\spP, where the Riemannian metric is one of zero curvature with isolated singularities, and is given by a holomorphic one-form ω on the surface. To this holomorphic one-form one can canonically associate a discrete subgroup Γ of PSL(2,\IR). If Γ happens to be a lattice (has cofinite volume), then it is known that all geodesic paths in the zero-curvature metric given by ω must either be closed or uniformly distributed in the surface X\spP. As a corollary, all billiard paths in the original polygon P must either be finite or uniformly distributed in P. A new class of examples of polygons P, whose associated group Γ is, in fact, a lattice have been discovered. At the same time, we have discovered the first examples of triangles P, as above, for which the associated groups Γ are not lattices (i.e. have infinite covolume). Finally, it is shown how to derive, in an explicit way, algebraic equations which specify the Riemann surface X\spP and one-form ω, which before were only described geometrically.