Integral transforms, such as the Laplace and Fourier transforms, have been major tools in mathematical physics for at least two centuries. In the last 25 years, the development of gauge theory has been closely related to new generalizations of integral transforms of a more geometric character. "Nahm and Fourier--Mukai Transforms in Geometry and Mathematical Physics" examines the differential-geometric constructions (Nahm) as well as the algebro-geometric approach (Fourier--Mukai functors). Also included is a considerable amount of material scattered in the literature and not systematically organized in any existing textbook or monograph. Key features: * Basic constructions and definitions are presented in preliminary background chapters so that the work is self-contained * Applications are presented * Open questions for graduate students and researchers * Good bibliography and index The book provides an introduction to current research in mathematical physics and is particularly useful to graduate students or researchers just entering this field.