Stochastic resetting occurs in a system that is returned to its initial state at a random sequence of times that is typically generated by a Poisson process with constant rate r. In stochastic search processes such as animal foraging and cellular transport, resetting has been posited as a mechanism for reducing the expected time to find a hidden target within some large search domain. In this chapter we consider various probabilistic formulations of diffusive search processes with stochastic resetting. We begin by showing how diffusion with instantaneous resetting can be represented as a jump-diffusion process. We construct a generalised Itô’s lemma and use this to derive density equations for a single particle and for a population of non-interacting particles. We then use the probabilistic formulation of instantaneous resetting to analyse the first passage time (FPT) problem for a single particle on the half-line with a partially absorbing target at the origin. Finally, we show how conditional expectations and renewal theory can be used to analyse the effects of finite return times and refractory periods on a diffusive search process with non-instantaneous resetting.