This book provides a comprehensive and penetrating account of the inverse Gaussian law. Beginning with an exhaustive historical overview that presents--for the first time--Etienne Halphen's pioneering wartime contributions, the book proceeds to a rigorous exposition of the theory of exponential families, focusing in particular on the inverse Gaussian law. The book also considers inverse natural exponential families and provides a detailed analysis of the "Tweedie" scale. A wealth of properties, characterization, new concepts of inverse exponential families, useful expositions of statistical results, and an updated list of about 400 key references are included as well. The book will be welcomed by students and researchers of statistics and probability.