A locally covariant renormalization group

Renormalization group flows, based on functional Polchinski or Wetterich equations, are powerful tools that give access to non-perturbative aspects of strongly coupled QFTs and gravity. I will provide an overview of a new approach, developed to construct a rigorous renormalization group (RG) flow on Lorentzian manifolds. This approach, based on a local and covariant regularization of the Wetterich equation, highlights its state dependence. I give the main ideas of a proof of local existence of solutions for the RG equation, when a suitable Local Potential Approximation is considered. The proof is based on an application of the renown Nash-Moser theorem. I will also present recent applications of the locally covariant RG equation to the non-perturbative renormalizability of quantum gravity.