Quantum aspects of 2+1 gravity

We review and systematize several recent attempts to canonically quantize general relativity in 2+1 dimensions, defined on space–times R×Σg, where Σg is a compact Riemann surface of genus g. The emphasis is on quantizations of the classical connection formulation, which use Wilson loops as their basic observables, but results from the ADM formulation are also summarized. We evaluate the progress and discuss the possible quantum (in)equivalence of the various approaches.