We present an overview of recent results for the classic problem of the survivalprobability of an immobile target in the presence of a single mobile trap or of acollection of uncorrelated mobile traps. The diffusion exponent of the traps is taken tobe either γ = 1, associated with normal diffusive motion, or0 < γ < 1, corresponding to subdiffusive motion. We considertraps that can only die upon interaction with the target and, alternatively, traps thatmay die due to an additional evanescence process even before hitting the target. Theevanescence reaction is found to completely modify the survival probability of the target.Such evanescence processes are important in systems where the addition of scavengermolecules may result in the removal of the majority species, or ones where the mobiletraps have a finite intrinsic lifetime.