This work develops an arithmetic analogue of the theory of ordinary differential equations, in which functions are replaced by integer numbers, the derivative operator is replaced by a Fermat quotient operator, and differential equations are replaced by arithmetic differential equations. The book partly follows a series of papers written by the author; however, a substantial part of the material presented here has never been published before. For most of the book, the only prerequisites are the basic facts of algebraic geometry and number theory. Author information is not given.