A linear semi-infinite program is an optimization problem with linear objective function and linear constraints in which either the number of unknowns or the number of constraints is finite. So linear semi-infinite optimization (or programming, LSIP in brief) can be seen as an extension of ordinary linear programming (LP) or, alternatively, as a particular branch of both semi-infinite (generally not even convex) optimization (SIP) and infinite-dimensional linear programming. Nevertheless, the many direct applications of LSIP, and its appealing theoretical properties, explain why LSIP became an independent active research area since its inception in the early 1960s, in a series of papers due to Charnes, Cooper and Kortanek. We knew these papers in the very late 1970s, together with work on the reduction approach in SIP done by Hettich and Jongen. We found SIP in general, and LSIP in particular, so enjoyable (maybe because of our geometrical temper) and so fascinating (for its many challenging open problems) that most of our research work during the last two decades has turned out to concentrate on this and related topics. It is our hope that this book will be able to at least partially communicate this feeling to the reader.