Free or interacting UV fixed points play a key role in the fundamental definition of QFT. In this talk, I give a broad overview of weakly and strongly interacting fixed points in 3d and 4d QFTs including models of particle physics with or without supersymmetry, and fermionic theories. Further, I explain methods and ideas to search for fixed points in 4d quantum gravity. Implications from the viewpoint of CFTs and higher-spin gauge theories through the AdS/CFT conjecture are also discussed.

It is commonly believed that for a quantum field theory to be fundamental, its high-energy behaviour must be governed by an ultraviolet (UV) fixed point. A very satisfying example is given by asymptotic freedom of QCD where the UV fixed point is non-interacting. In this talk, I discuss the existence of interacting UV fixed points in various quantum field theories, in particular in four dimensions, both with and without gravity. Implications for particle physics are also evaluated.