We consider billiards in "rectangular" polygons: polygons with the angles $\pi/2$ and $3\pi/2$. Changing the lengths of the sides of the polygon we obtain a family of billiard tables. We describe the enumerative geometry of closed trajectories of a.e. billiard in any such family.
We give asymptotical formulae for the number of closed billiard trajectories and for the number of generalized diagonals. The constants in these quadratic asymptotics are explicitly expressed in terms of the volumes of the corresponding strata in the moduli spaces of quadratic differentials on the Riemann sphere.
