Mathematical ideas introduced by Ramanujan a century ago in number theory and combinatorics have come to play a surprising role in understanding some deep and fundamental aspects of quantum gravity and quantum field theory in three very distinct contexts of holography, duality, and topology. In this colloquium, I shall first describe the fascinating history, physics, and mathematics behind this rich and fruitful connection focusing on the role (mock) modular forms have come to play in understanding quantum properties of black holes in string theory. I shall then elucidate briefly the manifestations of mock modularity in physics in its other avatars.

I will report on some recent progress on defining and computing finite size effects in the entropy of black holes with highly nontrivial agreements between thermodynamics and statistical mechanics. I will describe a number of puzzles and their resolutions along with some exact computations, and then briefly discuss the role of mock modular forms and Borcherds-Kac-Moody superalgebras in this context.