No one doubts that general relativity is an extremely accurate way of modeling even subtle aspects of of cosmos. No one doubts that quantum mechanics delivers accurate descriptions of nuclear and electromagnetic forces.
On the whole, general relativity is more troubled in practical applications. The gravitation constant is one of the the least precisely known of the fundamental physical constants. And, general relativity isn't sufficient in and of itself to explain observed cosmology. The notions of dark matter and dark energy must be introduced to resolve, even then, imperfectly, the discrepencies observed.
On the other hand, quantum mechanics uses a basically Newtonian conception of the nature of time space, modified by special relativity (called Minkowski space), but not by general relativity. We know that general relativity exists, and therefore know that the current model of quantum mechanics is incorrect.
One of the main approaches to reconciling general relativity and quantum physics is string theory, which is itself a generalization of the notion inherent in the gaps in the standard model of particle physics that two key particles, at least, are missing from the model. One is the graviton, which is analogous to the photon for the electromagentic force, the W and Z particles for the weak nuclear force, and the gluon for the strong nuclear force. The other is the Higgs boson, which would explain inertial mass. These models generally require far more than the usually 3+1 dimensions of space and time.
One of the leading schools of theoretical physics, other than string theory, is loop quantum gravity and some related theories. Unlike, string theory, in loop quantum gravity, general relativity effects arise from the nature of time-space itself, and there is no graviton, nor are there true singularities. Instead there is a quantization of space-time geometery itself. Black holes and the Big Bang are extreme phenomena in loop quantum gravity, but remain finite. The models suggest that a four dimensional time-space may be an "emergent" property of the underlying quantum geometrical rules.
The lack of singularities in LQG has spawned an article in a respected scientific journal discussing what this implies for the period before the Big Bang. The article's conclusion states:
As a general concept, this presents an attractive new universe picture, combining perpetual cyclic models and linear ones. A linear model starts at a finite time in the past, whereas a cyclic model exists forever, be it for several cycles or only for one contracting phase followed by an expanding one. We described a theory whose evolution never stops, being cyclic in this sense. But some traces of each cycle are irretrievably lost shortly after transiting from collapse to expansion. Complete predictions and explanations of observations can only be made for the finite part starting after the Big Bang. An eternal recurrence of the same is prevented by intrinsic cosmic forgetfulness. This may seem like a return to the traditional Big Bang picture, where speaking of 'before the Big Bang' is meaningless. But it is more subtle: in this traditional picture, the Big Bang is preceded by a singularity where the theory breaks down. The singularity is a theoretical limitation, rather than a physical beginning. Quantum gravity as used here can provide solutions extending through the classical singularity. And yet, limitations to observations of some, but not all, pre-Big Bang properties exist that are now derived within the theory, not as limitations to the theory.
In other words, in LQG, the Big Bang is not "the beginning" but merely the beginning of the current cycle of physical existence that wiped out most evidence of what came before it.
As an educated layman who has watched the debate play out for some time, I have to say that LQG looks more promising than string theory right now, despite the fact that string theory remains a majority position in the academy and is far better funded. LQG has made a great many strides in the last decade or so in refining what its theory can do. It isn't clear that string theory can break out of Minkowski space to solve its own problems as elegantly.
On the other hand, both string theory and LQG are rather short on testable predictions that can distinguish the two. Extrapolating from microscopic quantum phenomena to macroscopic observable phenomena is not a straight forward affair.
LQG stomps to death many of the pet theories of science fiction. Its lack of singularities makes notions like wormholes and extradimensions impossible. On the other hand, it is profound in another sense, with many versions of LQG seeing matter as essentially a contorted piece of time-space itself. It also isn't clear how LQG impacts some of the apparently non-local aspects of quantum mechanics (sometimes called, inaptly, quantum teleportation), that most defy a conventional causal notion of the nature of reality.