The monograph focuses on weighted functional spaces of the Sobolev type and their application to spectral theory and smoothness of solutions of differential equations with non-smooth coefficients . As compared to other monographs on the subject, emphasis is placated on non-smooth weights. This makes it possible to solve problems for a Schroedinger type operator with non-smooth potential. Norm and acceptable number estimates for the given embedding operator are in some cases final. These results lead to a generalization of the known criterion by Molchanov for spectrum discrimination and also to two-sided estimates of eigenvalues and to a criterion for nuclearity of the solution . The Otelbayev averaging is decribed in detail. This procedure makes it possible, in particular, to find the conditions under which B. Simon type inequalities are valid. The results of studying the solvability and separability of the nonlinear Sturm-Liouville equation determined by the localization method are presented.