This small booklet is based on lecture notes from a course in a special semester for advanced undergraduate students. It treats various mathematical disciplines connected with a description of an ideal (mathematical) billiard. In the book, connections to geometry and optics are stressed. Diverse methods are discussed and used (including symplectic geometry, calculus of variation, evolutes and involutes, a mathematical theory of rainbows, the Morse theory, 4-vertex theorem, and symplectic reduction). Various forms of billiards are studied (for instance completely integrable billiards, periodic billiards and dual billiards). The book is the first in a future series connected with the special semesters mentioned above. A good grounding in analysis, algebra, differential geometry and topology is helpful (but not necessary) for the reading of the book.