The main purpose of this work is construction of the mathematical theory of elastic plates and shells, by means of which the investigation of basic boundary value problems of the spatial theory of elasticity in the case of cylindrical domains reduces to the study of two-dimensional boundary value problems (BVP) of comparatively simple structure. In this respect in sections 2-5 after the introductory material, methods of re­ duction, known in the literature as usually being based on simplifying hypotheses, are studied. Here, in contradiction to classical methods, the problems, connected with construction of refined theories of anisotropic nonhomogeneous plates with variable thickness without the assumption of any physical and geometrical re­ strictions, are investigated. The comparative analysis of such reduction methods was carried out, and, in particular, in section 5, the following fact was established: the error transition, occuring with substitution of a two-dimensional model for the initial problem on the class of assumed solutions is restricted from below. Further, in section 6, Vekua's method of reduction, containing regular process of study of three-dimensional problem, is investigated. In this direction, the problems, connected with solvability, convergence of processes, and construction of effective algorithms of approximate solutions are studied.