Integral transforms, such as the Laplace and Fourier transforms, have been major tools in mathematical physics for at least two centuries. This book 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. The book provides an introduction to current research in mathematical physics and is particularly useful to graduate students or researchers just entering this field.