This book explores various topical trends in the theory of differentiable functions of several real variables and its applications. Among the subjects covered are: imbedding of various spaces of differentiable functions defined on sets in Euclidean space, on a sphere, and in a polydisc; approximation of functions; estimates for the norms of various integral operators in weighted space; conditions for stabilization of a function to a polynomial; sufficient conditions for multipliers; construction of unconditional bases in anisotropic spaces; existence of entire solutions for quasilinear equations; and establishment of an asymptotic formula for the kernels of powers of the resolvent of elliptic operators.