VOLUME A. Preface. History of Convexity (P.M. Gruber). Part 1: Classical Convexity. Characterizations of convex sets (P. Mani-Levitska). Mixed volumes (J.R. Sangwine-Yager). The standard isoperimetric theorem (G. Talenti). Stability of geometric inequalities (H. Groemer). Selected affine isoperimetric inequalities (E. Lutwak). Extremum problems for convex discs and polyhedra (A. Florian). Rigidity (R. Connelly). Convex surfaces, curvature and surface area measures (R. Schneider). The space of convex bodies (P.M. Gruber). Aspects of approximation of convex bodies (P.M. Gruber). Special convex bodies (E. Heil, H. Martini). Part 2: Combinatorial Aspects of Convexity. Helly, Radon, and Carathéodory type theorems (J. Eckhoff). Problems in discrete and combinatorial geometry (P. Schmitt). Combinatorial aspects of convex polytopes (M.M. Bayer, C.W. Lee). Polyhedral manifolds (U. Brehm, J.M. Wills). Oriented matroids (J. Bokowski). Algebraic geometry and convexity (G. Ewald). Mathematical programming and convex geometry (P. Gritzmann, V. Klee). Convexity and discrete optimization (R.E. Burkard). Geometric algorithms (H. Edelsbrunner). Author Index. Subject Index.