The book is devoted to the description and study of approximate methods of solving operator equations .. In the first part of the book (chapters 1-3), various iterative processes for solving linear and nonlinear equations are studied. Here, both traditional and essentially new conditions for the convergence of successive approximations are established; various a priori estimates of errors are derived; the influence of random errors (for example, the type of rounding errors) on convergence is analyzed; the distribution of errors is described, and t. d. Simple methods of obtaining estimates of spectral radii of linear operators are proposed, methods of approximate finding of own functions are studied. A new scheme for obtaining a posteriori estimates of errors. The second part of the book (chapter 4) contains a systematic construction of the theory of projection methods (Galerkin's method, Petrov's method, the method of moments, etc.) d. ) both in relation to the approximate solution of linear and nonlinear equations, and to the problem of eigenfunctions. Finally, in the last part (Chapter 5), approximate methods for solving nonlinear equations with parameters. are studied. The focus is on asymptotic approximations in the branching of solutions problem. A method for constructing asymptotic approximations to an implicit function. is proposed. The main text of the book significantly complements a large number of exercises of different difficulty.