Certain nonlinear optimisation problems arising in such disparate areas as the theory of computation, pure and applied probability and mathematical physics, can be solved by linear methods, provided one replaces the usual number system with one in which addition satisfies the idempotent law. Only recently has a systematic study of the subject emerged, triggered in part by a workshop organised by Hewlett-Packard’s Basic Research Institute in the Mathematical Sciences (BRIMS), which brought together for the first time many leading researchers in the area. This volume is a record of that workshop, but it also includes other invited contributions, a broad Introduction to Idempotency, written specially for the book, and a bibliography of the subject. In sum, this is the most up-to-date survey currently available of research in this developing area of mathematics; the articles cover both practical and more theoretical considerations, making it essential reading for all workers in the area.