The quest for a consistent theory of quantum gravity is one of the most important outstanding problems in theoretical physics. In the landscape of physical theories, quantum gravity sits at the corner where all the physical constants (speed of light, Newton’s and Planck’s constant) are finite. A region that is often overlooked is the nonrelativistic gravity regime. Contrary to common lore, it is becoming clear that the theory of nonrelativistic gravity is much richer than was so far appreciated, containing much more than just Newtonian gravity. Thus, this offers an entirely unexplored potential as a new route towards quantum gravity. Central to this development is the formulation of non-relativistic gravity in terms of Newton–Cartan type geometries, originally introduced by Cartan in 1923 to geometrize Newton's law of gravitation. I will review recent progress
that follows from a systematic expansion of general relativity in powers of 1/c, giving rise to a novel version of Newton-Cartan geometry. While being similar in spirit to the Post-Newtonian expansion, it has the advantage that it is covariant, off-shell and can be applied to any matter system coupled to GR. In the last part of the talk I wll address how this may help to address interactions between gravity and matter at the quantum level in the non-relativistic domain. Finally, I will briefly comment on approaches towards this, originating from non-relativistic limits of string theory and holography.