Abstract: Many supersymmetric CFTs possess a continuous space of vacua, known as a moduli space. A long-standing open problem is to determine whether a CFT has a moduli space using only its CFT data, without relying on a Lagrangian description. In this talk, we make progress on this question. First, assuming a continuous global symmetry is spontaneously broken on the moduli space, we show that the CFT must contain a tower of charged operators whose scaling dimensions grow linearly with the charge at large charge. Using intuition from holography and the weak gravity conjecture, we then propose and prove a bound on the leading correction to this linear behavior. Finally, without assuming a broken global symmetry, we use an analytic bootstrap equation to derive perturbative sum rules that constrain the data of any CFT with a moduli space.