The conventional lore is that quantum mechanics and gravity are not compatible, that there is a clash between these two fundamental ingredients of our world. This is considered one of the crucial problems in physics, and has spawned the field of Quantum Gravity.
However, the conventional view is wrong in some ways. Gravity and quantum mechanics in fact work fine together at ordinary energies. There are signs of trouble are extremely high energies – the Planck scale – but we can treat the two together in situations of low energy and low curvature such as our present world. This insight comes from effective field theory. Effective field theory is not a change in quantum theory, but is just a way of organizing the calculations in a way that separates the reliable part of the theory (low energy – which we have explored experimentally) from the high energy portion of the theory, which in gravity is completely unknown. Quantum calculation have the seemingly odd property of being sensitive to the unknown parts of the theory through loop integrals or sums over intermediate states, both of which probe all energies without bound. Effective field theory solves this problem by absorbing these unreliable contributions into the renormalization of parameters in a local Lagrangian, which then in principle should be determined by experiment. There remain reliable predictions that come from the low energy part of the theory. Gravity has worse than usual behavior at high energy and large curvature, which is the cause of the distress of quantum gravity. However, once one uses EFT to isolate the low energy effects, reliable calculations can be made within general relativity.
This viewpoint was first espoused (to my knowledge) by Steven Weinberg, who also taught us much about effective field theory. I was the first to do something along these lines. I had long realized that the quantum correction to the Newtonian gravitational potential was reliably calculable, but also knew that the result was too small to be detected. Eventually, I decided to do the calculation to demonstrate how the effective field theory treatment worked. This is published in the paper General relativity as and effective field theory: the leading quantum corrections . The full set of diagrams for the gravitational potential are calculated in work with Barry Holstein and Emil Bjerrum-Bohr, Quantum gravitational corrections to the non-relativistic scattering of two masses.
The Wikipedia article on quantum gravity actually starts with a reasonable explanation of the effective field theory treatment. For more advanced treatments, see reviews by Cliff Burgess and by myself.
Recently I have shown how the effective field theory can be reproduced in a dispersive approach, which does not require loop integrals. This helps explain the reliability of the method, as no high energy effects enter at all. I hope to write this up soon.
I remain interested in using the effective field theory treatment to understand the reliable aspects of quantum general relativity. I hope to do something soon on the issues of singularities. T. J. Blackburn has been discussing this with me.