General relativity has extremely solid foundations. Every experiment testing it has confirmed its predictions to great precision. For example, recent measurments of binary systems have confirmed the precise energy loss from the system that GR predicts from gravity wave radiation (in the Jupiter-Sun system the predicted loss of energy at a rate of a 40 watt bulb is impossible to measure accurately since the effect is so small). Predictions regarding gravitational impacts on the passage of time, the bending of light around massive objects, and frame dragging, for example, have also been confirmed with high preceision.
Like quantum chromodynamics (the mathematical physics of the strong force that binds atomic nuclei), the math of GR is not particularly easy to apply numerically in systems that are not highly stylized with assumptions include to simplify the calculations. It is fair to say that all of the mathematically easy cases of the implications of GR have been well explored, but that there are plenty of possible physical systems for which we do not have good numerical theoretical predictions - only approximations informed by analytical insights drawn from an understanding of the equations.
Thus, there are many theoretical predictions of GR that have not been experimentally tested because no one has done the math to make the prediction yet. In practice, a fair amount of the computer programs used to simulate gravitational mechanics at levels smaller than the entire universe and outside the immediate vicinity of very dense objects like black holes and neutron stars use a Newtonian gravity approximation.
The equations of quantum mechanics are formulated on a Minkowski background that includes special relativity but not general relativity. Usually this produces very little conflict - GR is generally applied to dynamics on a scale where quantum mechanical effects have little impact; QM is usually applied in circumstances where gravity is negligible relative to the effects of other forces. In the few circumstances where there is good reason to think that both GR and QM could have measurable impacts (e.g. in the early Big Bang and around black holes) there have been some ad hoc efforts by Bekenstein, Hawking and others to consider the effects of both, but there is no consensus means of formulating QM in a way that fully respects GR (or visa versa) in general.