  We review and extend in several directions recent results on the asymptotic
safety approach to quantum gravity. The central issue in this approach is the
search of a Fixed Point having suitable properties, and the tool that is used
is a type of Wilsonian renormalization group equation. We begin by discussing
various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff
procedure. We compare the beta functions of the gravitational couplings
obtained with different schemes, studying first the contribution of matter
fields and then the so-called Einstein-Hilbert truncation, where only the
cosmological constant and Newton's constant are retained. In this context we
make connection with old results, in particular we reproduce the results of the
epsilon expansion and the perturbative one loop divergences. We then apply the
Renormalization Group to higher derivative gravity. In the case of a general
action quadratic in curvature we recover, within certain approximations, the
known asymptotic freedom of the four-derivative terms, while Newton's constant
and the cosmological constant have a nontrivial fixed point. In the case of
actions that are polynomials in the scalar curvature of degree up to eight we
find that the theory has a fixed point with three UV-attractive directions, so
that the requirement of having a continuum limit constrains the couplings to
lie in a three-dimensional subspace, whose equation is explicitly given. We
emphasize throughout the difference between scheme-dependent and
scheme-independent results, and provide several examples of the fact that only
dimensionless couplings can have "universal" behavior.
