The Handbook has three aims. One is to survey, for experts, convex geometry in its ramifications and its relations with other areas of mathematics. A second aim is to give future researchers in convex geometry a high-level introduction to most branches of convexity and its applications, showing the major ideas, methods, and results; The third aim is to prove 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.