To answer this question, one must first understand how it came to be.
When it was discovered in the early twentieth century that Newtonian physics, although it had stood unchallenged for hundreds of years, failed to answer basic questions about time and space, such as 'Is the universe infinite?' or 'Is time eternal?', a new basis for physics was needed.
This lead to the development of Quantum Theory by Bohr, Schr'dinger and Heisenberg and Relativity Theory by Einstein. This was the first step in the development of a new basis for physics. Both theories, however are incomplete, and are limited in their abilities to answer many questions. Quantum Physics deals with the behaviour of very small objects, such as atoms, why they do not disintegrate as Newtonian Physics wanted. The theory of Relativity, on the other hand deals with much large scales, celestial bodies and others.
Both theories fail when confronted to the other's 'domain', and are therefore limited in their ability to describe the universe. One must unify these theories, make them compatible with one another. The resulting theory would be able to describe the behavior of the universe, from quarks and atoms to entire galaxies. This is the quantum theory of gravity.
Answered by:
Christian Kaas, M.A., Phyics Grad Student, IRNP, Paris

There are two fundamental areas of modern physics, each describes the universe on different scales. First we have quantum mechanics which talks about atoms, molecules and fundamental particles. Then we have general relativity which tells us that gravity is the bending and warping of space-time. There has been much work on finding a theory that combines these two pillars of physics.
There are three main aproches to quantum gravity all have there problems.
1) Loop quantum gravity.
2) String Theory.
3) Others; Penrose spin networks, Connes non-commutative geometry etc.
1) Loop quantum gravity is a way to quantise space time while keeping what General Relativity taught us. It is independent of a background gravitational field or metric. So it should be if we are dealing with gravity. Also, it is formulated in 4 dimensions. The main problem is that the other forces in nature, electromagnetic, strong and weak cannot be included in the formulation. Nor it is clear how loop quantum gravity is related to general relativity.
2) Then we have string theory. String theory is a quantum theory where the fundamental objects are one dimensional strings and not point like particles. String theory is "large enough" to include the standard model and includes gravity as a must. The problems are three fold, first the theory is background dependant. The theory is formulated with a background metric. Secondly no-one knows what the physical vacuum in string theory is, so it has no predictive powers. String theory must be formulated in 11 dimensions, what happened to the other 7 we cannot see? ( Also string theory is supersymmetric and predicts a load of new particles).
3) Then we have other approches, such as non-commutative geometry. This assumes that our space-time coordinates no longer commute. i.e. x y - y x is not zero. This formulation relies heavily on operator algebras.
All the theories have several things in common which are accepted as being part of quantum gravity at about Planck scale.
i)Space-time is discrete and non-commutative
ii)Holography and the Bekenstin bound.
i) This is "simply" applying quantum mechanics to space-time. In quantum mechanics all the physical observables are discrete.
ii) The holographic principle was first realised by Hawking. He realised that the entropy of a black hole was proportional to the surface area of the horizon and not the volume. That is all the information about a black hole is on the surface of the horizon. It is like a holograph, you only need to look at the 2-d surface to know everything you can about the black hole.
Bekenstin showed that there is a maximum amount of information that can pass through a surface. It is quantised in Planck units.
Answered by:
Andrew James Bruce, Physics Graduate, UK

'The mathematician's patterns, like the painter's or the poets, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.'