Two fundamental theories are commonly debated in the study of random processes: the Bachelier Wiener model of Brownian motion, which has been the subject of many books, and the Poisson process. While nearly every book mentions the Poisson process, most hurry past to more general point processes or to Markov chains. This comparative neglect is ill judged, and stems from a lack of perception of the real importance of the Poisson process. This distortion partly comes about from a restriction to one dimension, while the theory becomes more natural in more general contexts. This book attempts to redress the balance. It records the author's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions. The mathematical theory is powerful and a few key results often produce surprising consequences.