The book is the only systematic review of the theory of elliptic equations with partial derivatives in modern literature. The most important sections of the theory of linear and nonlinear elliptic equations of the second order are described in detail. The bibliography contains more than six hundred titles of works published mainly in the period 1924-1953. The book is designed primarily for mathematicians involved in differential equations. It is available to senior university students. In part, the book can be used by specialists involved in applications of the theory of differential equations, as well as related areas of mathematics.