Modified gravity theories may provide an alternative to dark energy to explain cosmic acceleration. We argue that the observational programme developed to test dark energy needs to be augmented to capture new tests of gravity on astrophysical scales. Several distinct signatures of gravity theories exist outside the ‘linear’ regime, especially owing to the screening mechanism that operates inside halos such as the Milky Way to ensure that gravity tests in the solar system are satisfied. This opens up several decades in length scale and classes of galaxies at low redshift that can be exploited by surveys. While theoretical work on models of gravity is in the early stages, we can already identify new regimes that cosmological surveys could target to test gravity. These include: (i) a small-scale component that focuses on the interior and vicinity of galaxy and cluster halos, (ii) spectroscopy of low-redshift galaxies, especially galaxies smaller than the Milky Way, in environments that range from voids to clusters, and (iii) a programme of combining lensing and dynamical information, from imaging and spectroscopic surveys, respectively, on the same (or statistically identical) sample of galaxies.