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 Ito and McKean."""