This book proposes a new mathematical methodology for addressing first passage problems, particularly in various classical stochastic models of applied probability.This approach is based on the so-called Abel-Gontcharoff (A-G) pseudopolynomials and the associated A-G expansions, which have been introduced and studied by the authors in recent years.These A-G expansions generalize the well-known Abel expansion, which allows us to extend the standard Taylor formula.Abel-Gontcharoff Pseudopolynomials and Stochastic Applications starts by presenting an in-depth presentation of the general theory, and then moves onto stochastic applications of this theory, especially in biomathematics.Univariate and multivariate versions of the A-G pseudopolynomials, as well as extensions with randomized parameters, are discussed and illustrated for modeling, notably by highlighting families of martingales and using stopping time theorems.This book concludes by paving the way to a nonhomogeneous theory for first crossing problems.