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This monograph is intended to be considered as my habilitation (D.Sc.)thesis; because of that and as everything has already appeared in English, itis performed exclusively in Russian.  The monograph comprises a detailed introduction and seven chapters thatrepresent part of my work influenced by Ap\'ery's proof from 1978 of theirrationality of $\zeta(2)$ and $\zeta(3)$, the values of Riemann's zetafunction. Chapter 1 is about "at least one of the four numbers $\zeta(5)$,$\zeta(7)$, $\zeta(9)$ and $\zeta(11)$ is irrational" (based in part onarXiv:math.NT/0206176). Chapter 2 explains a connection between the generalizedmultiple integrals introduced by Beukers in his proof of Ap\'ery's result andthe very-well-poised hypergeometric series; it is based onarXiv:math.CA/0206177. Chapter 3 surveys some arithmetic and hypergeometric$q$-analogies and establishes the irrationality measure$\mu(\zeta_q(2))<3.518876$ for a $q$-analogue of $\zeta(2)$; it closely followsthe text in Sb. Math. 193 (2002), 1151--1172, but also incorporates the sharperanalysis of the hypergeometric construction by Smet and Van Assche(arXiv:0809.2501 [math.CA]) to produce the improvement upon the 2002 result.Chapter 4 is devoted to the measure $\mu(\zeta(2))<5.095412$ and is based onarXiv:1310.1526 [math.NT]; Chapter 5 is establishing the estimate$||(3/2)^k||>0.5803^k$ for the distance from $(3/2)^k$ to the nearest integer,with the English version published in J. Th\'eor. Nombres Bordeaux 19 (2007),313--325. Chapter 6 reproduces the solution (from arXiv:math.CA/0311195) to theproblem of Asmus Schmidt about generalized Ap\'ery's numbers. Finally, Chapter7 is about expressing the special $L$-values as periods (in the sense ofKontsevich and Zagier), in particular, as values of hypergeometric functions;it is based on the publication in Springer Proc. Math. Stat. 43 (2013),381--395.

Ap\'ery's theorem and problems for the values of Riemann's zeta function and their $q$-analogues