Summary: Certain questions concerning the expansion of \(2\pi\)-periodic multi-variable functions in multiple Fourier series as well as estimates of remainders of some quadrature formulae are studied. It is known that in problems concerning convergence of these series and estimates of remainders of quadrature formulae, a significant role is played by the shift operator \(T_hf(x)=f(x+h)\) as well as by subsequently determined moduli of continuity of different orders. As for problems of the convergence of Fourier series in special functions of mathematical physics, analogous role play the generalized shift operators and further generated generalized modules of continuity. The latter are widely applied in the convergence of Fourier series of non-periodic functions both of one and many variables. Here, the generalized shift operators (Steklov operators) are applied in the problems of the convergence of multiple Fourier series. Estimates of the convergence speed of these series and exact and weak equivalent estimates of Kolmogorov's widths of some classes of functions are also given, as well as estimates of remainders of quadrature formalae in some classes of functions.