The object of this book is to study, from a mathematical standpoint, the problem of the equilibrium of a perfectly plastic body under certain conditions. This involves the solution of two problems in the calculus of variations: the strain problem and the stress problem. These problems have only recently been solved, and this volume aims to give an up-to-date account of this work. The tools and methods used should be useful in a number of fields, in particular in solving problems relating to the 'evolution' of certain plastic phenomena, the mechanics of fracture, and certain optimal control problems.