This book provides a survey of linear programming in semi-infinite and infinite-dimensional spaces. It includes a treatment of duality theory and of the fundamental theory of simplex-like algorithms for linear programs posed over vector spaces which may be infinite-dimensional. However, more than half the book is devoted to a detailed investigation of various types of infinite-dimensional linear program which occur, for example, in approximation theory, optimal control theory, dynamic networks, mass transfer problems and structural design. The approach is inductive; specific problems and algorithms are discussed in detail and the authors proceed from these to more general concepts and results.