This book, written by the founder of the stochastic calculus of variations, now a wide and prolific field of research, is divided into five independent parts that can each be read independently of the others. It is an account of the main recent developments in the field, taking care however over the geometric foundations, particularly the intrinsic point of view. Analytic applications are also presented. Included is the first presentation in book-form of quasi-sure analysis. §The book includes a vast bibliography of close to 800 items, and will be most valuable to all stochastic analysts. Parts of it can also be used for graduate courses.