Oriented matroids are a very natural mathematical concept which presents itself to us in many different guises, and which has connections and applications to many different areas. These areas include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science, and theoretical chemistry.

This book is intended for a diverse audience: graduate students who wish to learn the subject from scratch, researchers in the various fields of application who want to concentrate on certain aspects of the theory, specialists who need a thorough reference work, and all others between these extremes. There presently exists no comprehensive or accessible exposition of the field, and remedying this was our primary motivation for writing this book. It contains several new results that were developed in the course of its preparation, and we are confident that its appearance will stimulate many additional discoveries.

A list of problems and exercises is included after each of the ten chapters. These collections contain much information in addition to the main text. Unsolved problems are marked with a star. We wish to thank L. Billera, R. Bland, K. Brown, W. Fenton, K. Pukuda, J.E. Goodman, G. Gordon, T. Havel, V. Klee, J. Lawrence, C. Lee, R. Pollack, S. Sandvik, P. Shor, T. Terlaky, Th. Wanner, T. Zaslavsky, and numerous others who helped us at various stages of this project. We (or some subset of us) worked together on the manuscript at some time at each of our home institutions, and at IMA, RISC–Linz, Augsburg, DIMACS, Oberwolfach and the Mittag–Leffler Institute. We are grateful to all of these institutions for their support.