Generalized billiards (GB) describe a motion of a mass point (a particle) inside a closed domain formula_17 with the piece-wise smooth boundary formula_18. On the boundary formula_18 the velocity of point is transformed as the particle underwent the action of generalized billiard law. GB were introduced by Lev D. Pustyl'nikov in the general case, and, in the case when formula_20 is a parallelepiped in connection with the justification of the second law of thermodynamics. From the physical point of view, GB describe a gas consisting of finitely many particles moving in a vessel, while the walls of the vessel heat up or cool down. The essence of the generalization is the following. As the particle hits the boundary formula_18, its velocity transforms with the help of a given function formula_22, defined on the direct product formula_23 (where formula_24 is the real line, formula_25 is a point of the boundary and formula_26 is time), according to the following law. Suppose that the trajectory of the particle, which moves with the velocity formula_27, intersects formula_18 at the point formula_25 at time formula_30. Then at time formula_30 the particle acquires the velocity formula_32, as if it underwent an elastic push from the infinitely-heavy plane formula_33, which is tangent to formula_18 at the point formula_35, and at time formula_30 moves along the normal to formula_18 at formula_35 with the velocity formula_39. We emphasize that the "position" of the boundary itself is fixed, while its action upon the particle is defined through the function formula_40.