The AMS is excited to bring this volume, originally published in 1969, back into print. This well-written book has been used for many years to learn about stochastic integrals. The author starts with the presentation of Brownian motion, then deals with stochastic integrals and differentials, including the famous Ito lemma. The rest of the book is devoted to various topics of stochastic integral equations and stochastic integral equations on smooth manifolds. E. B. Dynkin wrote about the original edition in Mathematical Reviews: 'This little book is a brilliant introduction to an important boundary field between the theory of probability and differential equations'. These words continue to ring true today. This classic book is ideal for supplementary reading or independent study. It is suitable for graduate students and researchers interested in probability, stochastic processes, and their applications.