Mathematicians like to count things. Often in very complicated and fancy ways. In this talk I will explain how we can use quantum Airy structures — an abstract formalism recently proposed by Kontsevich and Soibelman, underlying the Eynard-Orantin topological recursion — to count various interesting geometric structures. Quantum Airy structures can be seen as a wide generalization of the famous Witten conjecture, connecting enumerative geometry, integrable systems, representation theory and mathematical physics. It is a great example of “physical mathematics” in action, with dualities in string theory and quantum field theory giving rise to fascinating, unexpected results in pure mathematics.