In this volume boundary value problems are studied from two points of view; solvability, unique or otherwise, and the effect of various smoothness properties of the given functions on the smoothness of the solutions. There are seven chapters contained in this volume. Chapter One gives a statement of the new results and an historical sketch. Chapter two introduces the various function spaces typical of modern Russian-style functional analysis. Chapters three and four deal with linear equations. Chapter six concerns itself with quasilinear equations, and chapter seven with systems of equations. These last four chapters can be read independently of one another.