The mathematical works of S.L.Sobolev were strongly motivated by particular problems coming from applications. In his celebrated book, 'Applications of Functional Analysis in Mathematical Physics', 1950, and other works, S.Sobolev introduced general methods that turned out to be very influential in the study of mathematical physics in the second half of the 20th century. This volume, dedicated to the centenary of S.L. Sobolev, presents the latest results on some important problems of mathematical physics, describing, in particular, phenomena of superconductivity with random fluctuations, wave propagation, perforated domains and bodies with defects of different types, spectral asymptotics for Dirac energy, Lamé system with residual stress, optimal control problems for partial differential equations and inverse problems admitting numerous interpretations. Methods of modern functional analysis are essentially used in the investigation of these problems.