  Invited contribution to the Encyclopedia of Mathematical Physics (2nd
edition), providing an overview over some main ideas and results in quantum
cosmology. Key points: Canonical quantisation of homogeneous, isotropic
cosmology; discussion of ambiguities in this quantisation; Construction of
explicit solutions, attempts at physical interpretation; conceptual issues
(time evolution, unitarity, probability interpretation); Introduction of scalar
fields or perfect fluids; Discussion of whether classical singularities are
resolved; Addition of inhomogeneities, possible predictions for primordial
cosmology; Connections to string theory and loop quantum gravity.
