Question: What is the current status of the problem of quantum gravity and the question regarding the reality of spacetime singularities?
About the article "General relativity", section "Current status".

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General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests.
However, there are strong indications the theory is incomplete.
The problem of quantum gravity and the question of the reality of spacetime singularities remain open.
Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.
Even taken as is, general relativity is rich with possibilities for further exploration.
Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations, and increasingly powerful computer simulations (such as those describing merging black holes) are run.
The race for the first direct detection of gravitational waves continues, in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.
More than ninety years after its publication, general relativity remains a highly active area of research.
The topology of spacetime is very complex and mathematical, but it can be summed up by saying that spacetime (at least from a general relativistic view) is globally and locally smooth, flat and continuous.
However, when one factors in quantum fluctuations (of the form formula_11), spacetime is not locally smooth.
Such areas as quantum gravity and string theory are trying to resolve this fundamental problem between quantum mechanics and general relativity. 
In general relativity, spacetime is assumed to be smooth and continuous—and not just in the mathematical sense.
In the theory of quantum mechanics, there is an inherent discreteness present in physics.
In attempting to reconcile these two theories, it is sometimes postulated that spacetime should be quantized at the very smallest scales.
Current theory is focused on the nature of spacetime at the Planck scale.
Causal sets, loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized spacetime with agreement on the order of magnitude.
Loop quantum gravity makes precise predictions about the geometry of spacetime at the Planck scale. 
The demand for consistency between a quantum description of matter and a geometric description of spacetime, as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.
Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.
At low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.
At very high energies, however, the result are models devoid of all predictive power ("non-renormalizability").
One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.
The theory promises to be a unified description of all particles and interactions, including gravity; the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.
In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.
Another approach starts with the canonical quantization procedures of quantum theory.
Using the initial-value-formulation of general relativity (cf.
evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined.
However, with the introduction of what are now known as Ashtekar variables, this leads to a promising model known as loop quantum gravity.
Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced, there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being dynamical triangulations, causal sets, twistor models or the path-integral based models of quantum cosmology.
All candidate theories still have major formal and conceptual problems to overcome.
They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available. 
Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result.
The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail.
Can quantum mechanics and general relativity be realized as a fully consistent theory (perhaps as a quantum field theory)?
Is spacetime fundamentally continuous or discrete?
Would a consistent theory involve a force mediated by a hypothetical graviton, or be a product of a discrete structure of spacetime itself (as in loop quantum gravity)?
Are there deviations from the predictions of general relativity at very small or very large scales or in other extreme circumstances that flow from a quantum gravity theory?The theory of LQG is one possible solution to the problem of quantum gravity, as is string theory.
There are substantial differences however.
For example, string theory also addresses unification, the understanding of all known forces and particles as manifestations of a single entity, by postulating extra dimensions and so-far unobserved additional particles and symmetries.
Contrary to this, LQG is based only on quantum theory and general relativity and its scope is limited to understanding the quantum aspects of the gravitational interaction.
On the other hand, the consequences of LQG are radical, because they fundamentally change the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum spacetime.
Presently, no semiclassical limit recovering general relativity has been shown to exist.
This means it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit (described by general relativity with possible quantum corrections).
Specifically, the dynamics of the theory is encoded in the Hamiltonian constraint, but there is no candidate Hamiltonian.
Other technical problems include finding off-shell closure of the constraint algebra and physical inner product vector space, coupling to matter fields of Quantum field theory, fate of the renormalization of the graviton in perturbation theory that lead to ultraviolet divergence beyond 2-loops (see One-loop Feynman diagram in Feynman diagram).
While there has been a recent proposal relating to observation of naked singularities, and doubly special relativity as a part of a program called loop quantum cosmology, there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity (a problem that plagues all current theories of quantum gravity).
Because of the above mentioned lack of a semiclassical limit, LQG has not yet even reproduced the predictions made by general relativity.
An alternative criticism is that general relativity may be an effective field theory, and therefore quantization ignores the fundamental degrees of freedom.