The functional renormalization group
is a tool in the systematic search for Euclidean QFTs that works with
very little input: All one needs to specify is a field content,
symmetries and a notion of locality. The functional renormalization
group then allows one to scan this theory space for bare actions for
which the path integral can be performed nonperturbatively. These
actions appear as fixed points (and relevant deformations) of the
renormalization group flow (so-called asymptotic safety). Such a
systematic search has so far not been performed for the tensor model
approach to quantum gravity. We investigate matrix models for
2-dimensional gravity and the Grosse-Wulkenhaar model, which is a simple
tensor model for a 4D noncommutative scalar field theory, as a first
step. I will report on work, done in collaboration with Astrid Eichhorn
and Alessandro Sfondrini, where we confirmed asymptotic safety purely
through functional RG methods. Based on the lessons learned from these
models I will summarize how the usual continuum RG approach has to be
generalized for the investigation of tensor models and conclude with a
recipe for the investigation of general tensor models.