Superconductivity is a striking phenomenon in which a system exhibits zero electrical resistance for temperatures below Tc, the transition temperature. The question of understanding whether there are quantum mechanical limits on Tc is thus one of great fundamental and technological importance.

I will start with a pedagogical introduction to the issues underlying this question, which should be easily accessible to students. I will then describe recent progress on deriving exact upper bounds on the Tc of 2D superconductors applicable to a wide range of quantum materials and to ultracold atomic gases. I will illustrate the usefulness of these bounds and make contact with experiments on a variety of strongly correlated superconductors where standard BCS theory fails. Next, I will show how these bounds need to be generalized when the band structure has either non-trivial topology or lacks dispersion, i.e., flat band superconductivity. I will conclude by discussing why the question of obtaining general upper bounds on Tc in 3D remains an open challenge.