  We give a pedagogical introduction to the basic ideas and concepts of the
Asymptotic Safety program in Quantum Einstein Gravity. Using the continuum
approach based upon the effective average action, we summarize the state of the
art of the field with a particular focus on the evidence supporting the
existence of the non-trivial renormalization group fixed point at the heart of
the construction. As an application, the multifractal structure of the emerging
space-times is discussed in detail. In particular, we compare the continuum
prediction for their spectral dimension with Monte Carlo data from the Causal
Dynamical Triangulation approach.
