Well, it's really about special relativity. General relativity is Newton's gravity incorporating the principle of special relativity. This waa Einsteins achoevrment. He accomplished it by showing that gravity was a field theory just like electromagnetism but where the field is the metric and spacetime the medium.
It's also understood how to incorporate special relativity into quantum theory. This was first accomplished by Feynman in his theory of QED. This was the quantum theory for the theory of electromagnetism. Later, similarctheories were established for the weak and strong force after Yang & Mills had written down their eponynous equations where they generalised the structure group of QED from U(1) to non-abelian groups. Later, Abdus Salam and Stephen Weinberg unified electromagnetism and the weak force into the electroweak force.
The main entry point into QED for calculations is through perturbative calculations (pQFT). Often it is claimed that QFT is perturbatively ill-defined. However, the mathematical technology has advanced far enough so that the perturbative framework is well-defined and rigorous. This is of course, as it should be, as pQFT was coming out with the correct results.
What remains is unifying QFTs with gravity. This means incorporating the relativised quantum theory with the relativised theoey of gravity. This is a long, uphill struggle. The main two contenders are Loop quantum gravity, which takes a conservative view and String Theory, which takes a radical view. And I mean radical. There are many objrcts in this theory which have not been observed - not only strings, but DBranes, higher dimensions and a tower of heavy states. Although it is often said, string theory has not given any testable predictions, there is actually one: higher curvature corrections to General Relativity. These have not, so far, been observed - or so I assume - otherwise the shouts of delight from the string theory community would have overwhelmed all of us!
The main entry point into quantum gravity is the semi-classical calculation of Hawking validating the Bekenstein-Hawking entropy formula for black holes. This has been derived in both the Loop Quantum Gravity frameworks by counting quantum geometry states and in String Theory, albeit in higher dimensions and with a higher number of charges.
This semiclassical calculation by Hawking uses the theory of QFT on curved spaces. It assumes that the dynamically curved spacetime changes slowly enough that it can be taken to be static. There is a well developed theory of how to do QFT on such backgrounds.