The pioneering work of Bekenstein and Hawking in the 70s showed that black
holes have a thermodynamic behavior. They produced a universal area law for
black hole entropy valid in the limit that the black hole is infinitely large.
Quantum effects induce finite-size corrections to this formula, thus providing a
window into the fundamental microscopic theory of gravity and its deviations
from classical general relativity. In this talk I will discuss recent advances in
high-precision computations of quantum black hole entropy in supersymmetric
theories of gravity, using new localization techniques. These calculations allow
us to test the suggestion that black holes are really ensembles of microscopic
states in a very detailed manner, much beyond the semi-classical limit.
I will then discuss how one can independently verify these calculations using
explicit models of microscopic ensembles for black holes in string theory
constructed in the 90s. These investigations throw up a surprising link to
number theory and the so-called Mock modular forms of Ramanujan. I will end
by sketching some research directions that these ideas lead to.