This second edition provides a comprehensive and unified presentation of the theory and applications of first-passage processes. It begins by developing the phenomenology from basic principles and illustrating it via the first-passage properties of the interval, the semi-infinite line, spherical geometries, unbounded domains, and bounded two-dimensional domains. Dedicated sections describe first-passage processes in many types of systems, including complete and nearly complete graphs, the Erdős-Renyi graph, the Cayley tree, and hierarchical networks. The new edition has been substantially revised and expanded to include new sections on the martingale method, heterogeneity, and other important concepts. A wide range of new applications are discussed, such as queueing, birth-death reactions, the Polya urn, resetting, chemoreception, search processes, and the 'hot hand' paradox. The book's accessible style makes it a valuable resource for researchers and students in statistical and mathematical physics, chemistry, engineering, and all other fields in which first-passage problems arise.