The semiclassical approximation to quantum gravity

A detailed review is given of the semiclassical approximation to quantum gravity in the canonical framework. This includes in particular the derivation of the functional Schr"odinger equation and a discussion of semiclassical time as well as the derivation of quantum gravitational correction terms to the Schr"odinger equation. These terms are used to calculate energy shifts for fields in De~Sitter space and non-unitary contributions in black hole evaporation. Emphasis is also put on the relevance of decoherence and correlations in semiclassical gravity. The back reaction of non-gravitational quantum fields onto the semiclassical background and the emergence of a Berry connection on superspace is also discussed in this framework.