We study the non-relativistic expansion of general relativity coupled to matter. Thisis done by expanding the metric and matter fields analytically in powersof 1/c2wherecis the speed of light. In order to perform this expansion it is shown tobe very con-venient to rewrite general relativity in terms of a timelike vielbein and aspatial metric.This expansion can be performed covariantly and off shell. We study the expansion ofthe Einstein–Hilbert action up to next-to-next-to-leading order.We couple this to dif-ferent forms of matter: point particles, perfect fluids, scalar fields (including an off-shellderivation of the Schr ̈odinger–Newton equation) and electrodynamics (both its electricand magnetic limits). We find that the role of matter is crucial in orderto understandthe properties of the Newton–Cartan geometry that emerges from the expansion of themetric. It turns out to be the matter that decides what type of clock form is allowed, i.e.whether we have absolute time or a global foliation of constant time hypersurfaces. Weend by studying a variety of solutions of non-relativistic gravity coupled to perfect fluids.This includes the Schwarzschild geometry, the Tolman–Oppenheimer–Volkoff solution fora fluid star, the FLRW cosmological solutions and anti-de Sitter spacetimes.