  We study the non-relativistic expansion of general relativity coupled to
matter. This is done by expanding the metric and matter fields analytically in
powers of $1/c^2$ where $c$ is the speed of light. In order to perform this
expansion it is shown to be very convenient to rewrite general relativity in
terms of a timelike vielbein and a spatial metric. This expansion can be
performed covariantly and off shell. We study the expansion of the
Einstein-Hilbert action up to next-to-next-to-leading order. We couple this to
different forms of matter: point particles, perfect fluids, scalar fields
(including an off-shell derivation of the Schr\"odinger-Newton equation) and
electrodynamics (both its electric and magnetic limits). We find that the role
of matter is crucial in order to understand the properties of the Newton-Cartan
geometry that emerges from the expansion of the metric. 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. We end by
studying a variety of solutions of non-relativistic gravity coupled to perfect
fluids. This includes the Schwarzschild geometry, the
Tolman-Oppenheimer-Volkoff solution for a fluid star, the FLRW cosmological
solutions and anti-de Sitter spacetimes.
