This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with Analys . . . . .
The survey consists of five chapters: the first provides some general background; the second
concerns convex smooth billiards (elliptic case); the third deals with billiards in polygons and polyhed . . . . .