This book aims to provide a unique introduction to the mathematics of computing with imperfect information. This rapidly developing field has been represented by a growing number of diverse contributions to academic journals and the proceedings of international conferences. The aim here is to bring a new coherence and clarity to this complex and expanding body of research, and to give a comprehensive overview that will be invaluable to both experts and research students new to the area.In many real-world contexts, it is necessary to make decisions based on incomplete information. This book deals with the mathematics underlying the practical computations that it entails. The theory impacts on a wide range of fields, within mathematics and beyond, and has given rise to unexpected connections and deep and interesting ongoing research. This book is essential reading for mathematicians and computer scientists concerned with the computational structure of information, and many others for whom computing in a real context involves working on emergent data.