  This is a review article on quantum gravity. In section 1, the Penrose
singularity theorem is proven. In section 2, the covariant quantization
approach of gravity is reviewed. In section 3, an article by Hawking is
reviewed that shows the gravitational path integral at one loop level to be
dominated by contributions from some kind of virtual gravitational instantons.
In section 4, the canonical, non-perturbative quantization approach is
reviewed. In section 5, arguments from Hawking are mentioned which show the
gravitational path integral to be an approximate solution of the Wheeler deWitt
equation. In section 6, the black hole entropy is derived in various ways.
Section 6.1 uses the gravitational path integral for this calculation. Section
6.2 shows how the black hole entropy can be derived from canonical quantum
gravity. In section 7.1, arguments from Dvali and Gomez who claim that gravity
can be quantized in a way which would be in some sense self-complete are
critically assessed. In section 7.2 a model from Dvali and Gomez for the
description of quantum mechanical black holes is critically assessed and
compared with the standard quantization methods of gravity.
