The book describes in detail the birational geometry of algebraic tori, a number of problems in the theory of integer representations of finite groups that arose in connection with the geometry of tori are considered in detail. An important place is occupied by the study of the connections of the birational properties of tori with their arithmetic (Tamagawa numbers, approximation problems, Minkowski-Hasse problem, etc.). The last chapter deals with the problem of the rationality of the fields of invariants of finite abelian groups. The monograph is quite autonomous, the first three chapters contain all the auxiliary material necessary for further from the theory of group schemes.