*Abstract:* A localized structure is a solution of a partial differential equation that is close to a `trivial' equilibrium state, except in a number of `small' spatial regions where the solution is `far-from-equilibrium'. In the context of ecology, such structures may represent vegetated area's embedded in bare soil: the so-called vegetation patterns that play a central role in the process leading to desertification. In polymer electrolyte membrane (PEM) fuel cells, networks of thin localized structures are crucial for the transport of electrical currents. The mathematical models that exhibit these structures may have very distinct characters: vegetation patterns are governed by systems of reaction-diffusion equations, the polymer-dynamics minimize Cahn-Hilliard type energies -- leading to scalar PDEs with up to sixth order spatial derivatives. In the talk, it will be shown how interactions between mathematicians and ecologists or chemists induce fundamental questions about the mathematical theory of localized structures. In the second half of the talk, novel mathematical insights obtained on the bifurcation scenario's of pulses that have been inspired by ecology will be discussed.