One aim of this handbook is to survey convex geometry, its many ramifications and its relations with other areas of mathematics. As such it should be a useful tool for the expert. A second aim is to give a high-level introduction to most branches of convexity and its applications, showing the major ideas, methods and results. This aspect should make it a source of inspiration for future researchers in convex geometry. The handbook should be useful for mathematicians working in other areas, as well as for econometrists, computer scientists, crystallographers, physicists and engineers who are looking for geometric tools for their own work. In particular, mathematicians specializing in optimization, functional analysis, number theory, probability theory, the calculus of variations and all branches of geometry should profit from this handbook.