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Closed billiard trajectories for polygonal billiards have been studied a lot. The best results have been achieved for polygons which satisfy the so called Veech alternative: for every direction, either all, or no, billiard trajectories are closed. In particular, this holds for all regular polygons. Our recent work with Diana Davis and Serge Tabachnikov contains results regarding the lengths (both geometric and combinatorial) of closed billiard trajectories in regular polygons, and also regarding their symbolic orbits (sequences of sides of reflection). In the geometrically significant case of a regular pentagon our results are more or less complete. We have partial results (both rigorously proved and experimental) for other regular polygons. No preliminary knowledge is required.