We present a nonperturbative quantization of general relativity coupled to dust and other matter fields. The dust provides a natural time variable, leading to a physical Hamiltonian with spatial diffeomorphism symmetry. The surprising feature is that the Hamiltonian is not a square root. This property, together with the kinematical structure of loop quantum gravity, provides a complete theory of quantum gravity, and puts applications to cosmology, quantum gravitational collapse, and Hawking radiation within technical reach.

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2+1 Einstein gravity is used as a toy model for testing a program for nonperturbative canonical quantisation of the 3+1 theory. The program can be successfully implemented in the model and leads to a surprisingly rich quantum theory.

We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity programme, and is based on an alternative to the Schrödinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularity is avoided in the quantum dynamics. We discuss these results with a view to identifying the criteria that constitute "singularity resolution" in quantum gravity.

We give an exact spherically symmetric solution for the Einstein-scalar field system. The solution may be interpreted as an inhomogeneous dynamical scalar field cosmology. The spacetime has a timelike conformal Killing vector field and is asymptotically conformally flat. It also has black or white holelike regions containing trapped surfaces. We describe the properties of the apparent horizon and comment on the relevance of the solution to the recently discovered critical behaviour in scalar field collapse.