The averaged null energy condition, or ANEC, is a fundamental bound on negative energy in quantum field theory. It has a surprising array of applications, from black holes to quantum information to the 3D Ising model, and often seems to be the crucial ingredient to ensure that interacting quantum field theories are consistent. In this talk, I will discuss the role of the ANEC in understanding the general structure of the renormalization group in two and four dimensions. The main results are that the averaged null energy defines a natural metric on the space of field theories, and the ANEC implies that the renormalization group is monotonic.