The theory of approximation of functions is one of the central branches in mathematical analysis and has been developed over a number of decades. This monograph deals with a series of problems related to one of the directions of the theory, namely, the approximation of periodic functions by trigonometric polynomials generated by linear methods of summation of Fourier series. More specificly, the following linear methods are investigated: classical methods of Fourier, Fejer, Riesz, and Roginski. For these methods the so-called Kolmogorov-Nikol'skii problem is considered, which consists of finding exact and asymptotically exact qualities for the upper bounds of deviations of polynomials generated by given linear methods on given classes of 2-periodic functions. Much attention is also given to the multidimensional case.