Classically the first time a particle reaches a target, either via a diffusive mechanism or deterministically, controls many processes in science. In the absence of a well defined path the first arrival time to a target state of a quantum particle can be treated in several ways. Such problems arise in the excitation transfer to a reaction centre in light harvesting systems, and more recently in the context of quantum search algorithms. We will review the challenges of search, starting with the quantum renewal equation, dark states and their relation to symmetry, and topological aspects of the recurrence problem. Exploiting quantum dynamics pierced by measurements we study these effects for simple quantum walks on a graph and for many-body spin systems.