The title of this lecture can be understood in two ways. Literally, in a somewhat facetious sense: mathematicians are playing by launching billiard balls on tables of various forms and observing (and also trying to predict) what happens. In a more serious sense, the expression “playground” should be understood as “testing area”: various questions, conjectures, methods of solution, etc. in the theory of dynamical systems are “tested” on various types of billiard problems. I hope to demonstrate convincingly that at least the second interpretation deserves serious attention.

The literature concerning billiards is rather large, including scientific papers as well as monographs, textbooks, and popular literature. Short brochures by G. A. Galperin and A. N. Zemlyakov and by G. A. Galperin and N. I. Chernov are written in a rather accessible manner, and touch a broad circle of questions. An introduction to problems related with billiards for a more advanced reader is contained in Chapter 6 of the book. The next level is represented by a very well written book of S. Tabachnikov, whose publication in Russian is unfortunately delayed. The book by the author and B. Hasselblatt contains a rather detailed modern exposition of the theory of convex billiards and twisting maps. A serious but rather accessible exposition of modern state of the theory of parabolic billiards is contained in a survey paper by H. Masur and S. Tabachnikov which will be published (in English) in spring 2002. The collection of papers contains rich material on hyperbolic billiards and related questions. More special references will be given below during the exposition.