Geometric constraints affect pattern selection and topological defect formation in a wide range of non-equilibrium processes, from crystal growth to morphogenesis. In the first part of this talk, I will summarize recent experimental and theoretical work that aims to understand how curvature controls symmetry breaking and topological defects on the wrinkled surfaces of elastic bilayer materials [1]. Specifically, we will present a higher-order PDE model that captures essential characteristics of the experimental data. In the second part, we will generalize the underlying ideas  to obtain an analytically tractable description [2] of bacterial and other active suspensions. The resulting generalized Navier-Stokes equations reveal an unexpected chiral symmetry-breaking mechanism, and offer insight into the triad dynamics of classical turbulence by uncovering a previously unknown cubic invariant [3].