Systems with multiple timescales, in which the dynamics separate into slow and fast components, occur ubiquitously in models of physical, biological and ecological processes. In this talk I will focus on two example applications: the dynamics of vegetation patterns in water-limited regions and the propagation of impulses along nerve fibers. These processes can be modeled by reaction diffusion PDEs in which the patterns arise as traveling wave solutions. The slow/fast timescale separation induces a geometry on the underlying equations which can be exploited to gain insight into the structure and stability of these patterns.