Abstract: Encounter-based methods provide a general probabilistic framework for modelling absorption on the surface or interior of a target. An absorption event occurs when the contact time with the target exceeds a random threshold. If the probability distribution of the latter is an exponential function, then one recovers the Markovian example of absorption at a constant rate, whereas a non-exponential distribution signifies non-Markovian absorption. In the case of a partially absorbing target interior (surface) the contact time is given by a Brownian functional known as the occupation time (boundary local time). In this talk we review encounter-based methods for diffusive search processes, and then show how to extend the theory to search processes with stochastic resetting, anomalous diffusion, and active run-and-tumble particles.