This volume contains survey articles based on the invited lectures given at the Twentieth British Combinatorial Conference, organized jointly by the University of Durham and the Open University. It was held in July 2005 at the University of Durham. This biennial conference is a well-established international event, with speakers from all over the world. By its nature this volume provides an up-to-date overview of current research activity in several areas of combinatorics, ranging from combinatorial number theory to geometry. The authors are some of the world's foremost researchers in their fields.