  The methods of the renormalization group and the $\varepsilon$-expansion are
applied to quantum gravity revealing the existence of an asymptotically safe
fixed point in spacetime dimensions higher than two. To facilitate this,
physical renormalization schemes are exploited where the renormalization group
flow equations take a form which is independent of the parameterisation of the
physical degrees of freedom (i.e. the gauge fixing condition and the choice of
field variables). Instead the flow equation depends on the anomalous dimensions
of reference observables. In the presence of spacetime boundaries we find that
the required balance between the Einstein-Hilbert action and
Gibbons-Hawking-York boundary term is preserved by the beta functions.
Exploiting the $\varepsilon$-expansion near two dimensions we consider Einstein
gravity coupled to matter. Scheme independence is generically obscured by the
loop-expansion due to breaking of two-dimensional Weyl invariance. In schemes
which preserve two-dimensional Weyl invariance we avoid the loop expansion and
find a unique ultra-violet (UV) fixed point. At this fixed point the anomalous
dimensions are large and one must resum all loop orders to obtain the critical
exponents. Performing the resummation a set of universal scaling dimensions are
found. These scaling dimensions show that only a finite number of matter
interactions are relevant. This is a strong indication that quantum gravity is
renormalizable.
