  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+\eps 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'. Further it is argued that as a consequence of the
scenario the selfinteractions 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.
