The displacement of one fluid by another in a porous medium gives rise to a rich variety of hydrodynamic instabilities. Beyond their scientific value as fascinating models of pattern formation, unstable porous-media flows are essential to understanding many natural and man-made processes, including water infiltration in the vadose zone, carbon dioxide injection and storage in deep saline aquifers, methane venting from organic-rich sediments, and fracturing from fluid injection. Here, we review a handful of these hydromechanical instabilities, elucidate the key physics at play, and point to modeling frameworks that permit quantitative assessments of their impact at the geologic scale.