  On the basis of a limited number of reasonable axioms, we discuss the
classification of all the possible universality classes of diffeomorphisms
invariant metric theories of quantum gravity. We use the language of the
renormalization group and adopt several ideas which originate in the context of
statistical mechanics and quantum field theory. Our discussion leads to several
ideas that could affect the status of the asymptotic safety conjecture of
quantum gravity and give universal arguments towards its proof.
