Recent progress in the theory of (super)gravity has made it possible to calculate the entropy of certain black holes beyond the area law of Bekenstein and Hawking in a large charge expansion. I will first review this in the context of string theory. I will then present my own recent and ongoing work on the following interrelated questions:

1. How to compute and understand exponentially suppressed contributions to the black hole entropy?
2. How to compute the density of states of a conformal field theory in a regime far away from the Cardy regime?
3. How does one understand the entropy when there is an issue of wall-crossing in the moduli space?

The answers involve an interplay of physics and mathematics - in particular, newly discovered number theoretic objects called mock modular forms.