The asymptotic safety scenario in quantum gravity: an introduction

The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. All presently known evidence is surveyed: ( a) from the 2 + epsilon expansion, ( b) from the perturbation theory of higher derivative gravity theories and a 'large N' expansion in the number of matter fields, ( c) from the 2- Killing vector reduction and ( d) from truncated flow equations for the effective average action. Special emphasis is given to the role of perturbation theory as a guide to 'asymptotic safety'. Furthermore it is argued that as a consequence of the scenario the self-interactions appear two dimensional in the extreme ultraviolet. Two appendices discuss the distinct roles of the ultraviolet renormalization in perturbation theory and in the flow equation formalism.