Arens-Eells spaces (aka Lipschitz free spaces or transportation cost spaces) give rise to interesting examples of Banach spaces and provide analytic techniques within Banach space geometry. But they are also of importance as a tool for analysing objects outside Banach space theory using functional analytical techniques. I will present two such uses. The first is to abstract harmonic analysis where Arens-Eells spaces can be used to provide a very simple conceptual proof of a recent characterisation of amenability of topological groups due to F. M. Schneider and A. Thom. The second application is to the geometric study of topological groups, namely, to establish the Gromov correspondence between coarse equivalence and topological couplings in the widest possible setting.