This study reviews the history of Newton-Cartan (NC) 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 leads to either NC geometry with absolute time or NC geometry with non-relativistic gravitational time dilation effects. The latter shows that non-relativistic gravity (NRG) includes a strong field regime and goes beyond Newtonian gravity. We start by reviewing early developments in NC geometry, including the covariant description of Newtonian gravity, mainly through the works of Trautman, Dautcourt, Kunzle, and Ehlers. We then turn to more modern developments, such as the gauging of the Bargmann algebra and 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 NC geometry. Finally, we discuss matter couplings, solutions, and odd powers in 1/c and conclude with a brief summary of related topics.