The geometry of the moduli space of 4d moduli spaces, and in particular of their Coulomb branches (CBs), is very constrained. In this talk I will show that through its careful study, we can learn general and somewhat surprising lessons about the properties of super conformal field theories (SCFTs). Specifically I will show that we can prove that the scaling dimension of CB coordinates, and thus of the corresponding operator at the SCFT fixed point, has to be rational and it has a rank-dependent maximum value and that in general the moduli spaces of SCFTs can have metric singularities as well as complex structure singularities.
Finally I will outline how we can explicitly perform a classification of geometries of SCFTs and carry out the program up to rank-2. The results are surprising and exciting in many ways.