A polygon is called rational if the angle between each pair of sides is a rational multiple of $\pi$. The main theorem we will prove is

Theorem 1:  For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has "many" periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $\textbf{S}^1$.

We will also prove some refinements of Theorem 1: the "well distribution" of periodic orbits in the polygon and the residuality of the points $q \in Q$ with a dense set of periodic directions.