Markov Process -- Basic Properties -- Hunt Process -- Brownian Motion -- Potential Developments -- Generalities -- Markov Chains: a Fireside Chat -- Ray Processes -- Application to Markov Chains -- Time Reversal -- h-Transforms -- Death and Transfiguration: A Fireside Chat -- Processes in Duality -- The Martin Boundary -- The Basis of Duality: A Fireside Chat.From the reviews of the First Edition: "This excellent book is based on several sets of lecture notes written over a decade and has its origin in a one-semester course given by the author at the ETH, Zürich, in the spring of 1970. The author's aim was to present some of the best features of Markov processes and, in particular, of Brownian motion with a minimum of prerequisites and technicalities. The reader who becomes acquainted with the volume cannot but agree with the reviewer that the author was very successful in accomplishing this goal…The volume is very useful for people who wish to learn Markov processes but it seems to the reviewer that it is also of great interest to specialists in this area who could derive much stimulus from it. One can be convinced that it will receive wide circulation." (Mathematical Reviews) This new edition contains 9 new chapters which include new exercises, references, and multiple corrections throughout the original text.