This is a fourth edition of the classic text "A Handbook of Stochastic Methods" which has been significantly augmented, thoroughly revised, and restructured to accomodate the new material within a systematic logical framework. This new edition adheres the original aim: "to make available in simple language and deductive form, the many formulae and methods that can be found in the literature on stochastic methods." A new chaper on the applications of stochastic methods in finance provides an introduction to this field using the same simple kind of language as the other parts of the book. This chapter also includesn introduction to Lévy processes, which have found to be very useful in simulating financial systems, where more accuracy is required than is available from simple Brownian motion models. New material is also provided on the approach to the white noise limit, on the applications of Poisson representation methods to population dynamics, and on several other applications of stochastic methods.