This book presents the theory of functions spaces, now known as Sobolev spaces, which are widely used in the theory of partial differential equations, mathematical physics, and numerous applications. The author also treats the variational method of solution of boundary value problems for elliptic equations, including those with boundary conditions given on manifolds of different dimensions. In addition, the theory of the Cauchy problem for second-order hyperbolic equations with variable coefficients is studied. The book is intended for researchers in mathematics and mathematical physics and would be useful to undergraduate and graduate students taking advanced courses in these areas.