This paper surveys recent results involving boundary crossing probabilities and related statistical applications. The first part is concerned with problems of sequential analysis, especially repeated significance tests and their application to sequential clinical trials involving survival data. The second part develops the probability theory motivated by the problems of Part 1. A method for computing first passage distributions of Brownian motion to linear boundaries is introduced and then modified to handle problems problems in discrete time and those involving nonlinear boundaries. The third part is concerned with fixed sample statistical problems, especially change-point problems, which involve boundary crossing probabilities. Examples are given of problems for which the methods of Part 2 appear adequate and of problems which require new methods.