  We review the history of Newton-Cartan gravity with an emphasis on recent
developments, including the covariant, off-shell large speed of light expansion
of general relativity. Depending on the matter content, this expansion either
leads to Newton-Cartan geometry with absolute time or to Newton-Cartan geometry
with non-relativistic gravitational time dilation effects. The latter shows
that non-relativistic gravity includes a strong field regime and goes beyond
Newtonian gravity. We start by reviewing early developments in Newton-Cartan
geometry, including the covariant description of Newtonian gravity, mainly
through the works of Trautman, Dautcourt, K\"unzle and Ehlers. We then turn to
more modern developments, such as the gauging of the Bargmann algebra, and we
describe why the latter cannot be used to find an off-shell covariant
description of Newtonian gravity. We review recent work on the $1/c$ expansion
of general relativity and show that this leads to an alternative `type II'
notion of Newton-Cartan geometry. Finally, we discuss matter couplings,
solutions and odd powers in $1/c$, and we conclude with a brief summary of
related topics.
