This book is devoted to boundary value problems of infinite order and the corresponding Sobolev spaces. Chapter I covers the necessary and sufficient conditions of nontriviality in three of the most commonly encountered cases of analysis: the bounded region, full Euclidean space and the torus. In Chapter II, the nonlinear elliptic Dirichlet problem is studied. A stochastic problem of the theory of elasticity is considered as an application. One of the aims of Chapter III is to study the nonhomogeneous Dirichlet problem of infinite order. In order to do this, the `trace' theory of Sobolev spaces is constructed. Chapter IV develops a general views of Sobolev spaces with their conjugates as limits of Banach spaces. In Chapter V some results of the imbeddingtheory of Sobolev spaces of infinite order are otained. In Chapter VI the solvability of the mian boundary value problems for parabolic and hyperbolic equations of infinite order is established.