Since its first publication in 1965 in the series Grundlehren der mathematischen Wissenschaften this book has had a profound and enduring influence on research into the stochastic processes associated with diffusion phenomena. Generations of mathematicians have appreciated the clarity of the descriptions given of one- or more- dimensional diffusion processes and the mathematical insight provided into Brownian motion. Now, with its republication in the Classics in Mathematics it is hoped that a new generation will be able to enjoy the classic text of Itô and McKean. TOC:The standard Brownian motion. Brownian local times. The general 1-dimensional diffusion. Generators. Time changes and killing. Local and inverse local times. Browinian motion in several dimensions. A general view of diffusion in several dimentions. Bibliography. List of notations. Index