Black holes and other compact objects can be described at long distances by a point particle effective field theory. In such theory, tidal effects and the dynamics of the horizon are captured in a series of Wilson coefficients which can be determined by matching with a complete description of the black hole in general relativity. Surprisingly, even in the classical theory these coefficients undergo a renomalization group flow, which arises from an ambiguity in separating the black hole from its environment. In this talk I will explain the basic setup of this EFT and a systematic procedure for carrying out the matching and running of its Wilson coefficients using scattering amplitudes in the black hole background and the EFT. Time permitting, I will describe hints of a surprising behavior in the EFT describing extremal black holes in maximal supergravity.