This 2007 volume contains survey articles based on the invited lectures given at the Twenty-first British Combinatorial Conference held in July 2007 at the University of Reading. This biennial conference is a well-established international event and the articles are of the high quality that befits the event. By its nature this volume provides an overview of current research activity in several areas of combinatorics ranging from graph theory to current applications of combinatorial mathematics including efficient approximability of NP-hard optimization problems and cryptographic key management. The authors are some of the world s foremost researchers in their fields and here they summarize existing results and give a unique preview of work currently being written up. The book provides a valuable survey of the state of knowledge in combinatorics. It will be useful to research workers and advanced graduate students primarily in mathematics but also in computer science statistics and engineering.