Applications of first passage times in stochastic processes arise across a wide range of length and time scales in biological settings. After an initial technical overview, we survey representative applications and their corresponding models. Within models that are effectively Markovian, we discuss canonical examples of first passage problems spanning applications to molecular dissociation and self-assembly, molecular search, transcription and translation, neuronal spiking, cellular mutation and disease, and organismic evolution and population dynamics. In this last application, a simple model for stem-cell ageing is presented and some results derived. Various approximation methods and the physical and mathematical subtleties that arise in the chosen applications are also discussed.