Manifolds are important entities in dynamical systems, and organise space into regions in which different motions occur. For example, intersections between stable and unstable manifolds in discrete systems result in chaotic motion. This talk will focus on manifolds and their locations in continuous dynamical systems, and in particular on Melnikov's method and its adaptations for determining the effect of perturbations on manifolds. The relevance of such adaptations to a surprising range of applications will be shown, in addition to recent theoretical developments inspired by such problems. The applications addressed in this talk include understanding the motion of fluid near oceanic eddies and currents, optimising mixing in nano-fluidic devices in order to improve reactions, computing the speed of a flame front, and finding the spreading rate of bacterial colonies.