The monograph is devoted to a systematic exposition of the results of studying the approximate characteristics of periodic functions of several variables from the Besov classes \(B_{p,\theta}^r\) which are determined by the dominant mixed smoothness. The book also discusses results in the field of approximation of functions from the Sobolev classes \(W_{p,\alpha}^r\) defined by restrictions on mixed derivative and approximation of functions from the Nikol’skiĭ classes \(H_{p}^r\) defined by restrictions on the corresponding differences. In the first chapter problems of approximation of functions from \(B_{p,\theta}^r\) classes by trigonometric polynomials are investigated. The main attention is focused on finding exact order estimates for approximation of functions from these classes by step hyperbolic Fourier sums and also on the best approximations by trigonometric polynomials with the corresponding spectrum. Considerable attention in this chapter is also paid to studying bounded linear operators, which ensure the best order approximations of functions from the classes \(B_{1,\theta}^r\) in the \(L_1\) metric and functions from the classes \(B_{\infty,\theta}^r\) in the uniform metric. The second and third chapters of the book are devoted to the non-linear approximation of functions from \(B_{p,\theta}^r\) classes and in some cases from \(W_{p,\alpha}^r\) and \(H_{p}^r\) classes. These are the so-called best and orthogonal \(M\)-term trigonometric approximations. In addition, a significant part of the third chapter is given to results of studying of approximations of functions of \(2d\) variables from \(B_{p,\theta}^r\) classes by linear combinations of products of functions of \(d\) variables. These approximations are called bilinear. The corresponding approximate characteristics are closely connected with the best \(M\)-term trigonometric approximations, as well as with the Kolmogorov widths of the corresponding classes of functions. In the final, fourth chapter, properties of a number of widths of functional classes (Kolmogorov, linear, trigonometric and orthogonal) is investigated. Results presented here, on the one hand, are of independent interest and, on the other hand, they serve as justification for the approximation of functions from the classes \(B_{p,\theta}^r\) by trigonometric polynomials with ‘numbers’ of harmonics from the step hyperbolic crosses.