After revising the asymptotic safety conjecture for quantum gravity, I will present how the concept of universality can be employed to classify the possible ultraviolet completions of general relativity. In particular, we will focus on quantum gravity close to two dimensions, and discuss how different realizations of the diffeomorphisms group, though locally isomorphic, give rise to different quantum theories. We will finally speculate on the analytic continuation of such models far from the two-dimensional case and on the existence of an ultraviolet fixed point for quantum gravity in four dimensions.