Résumé
This series of lectures aims to explain remarkable (mock) modular properties of generating functions of certain BPS indices which, on the one hand, coincide with generalized Donaldson-Thomas invariants of Calabi-Yau (CY) threefolds and, on the other hand, count the entropy of supersymmetric black holes appearing in compactifications of Type II string theory on these manifolds. These properties can be used to explicitly compute the generating functions up to a finite number of parameters. In the case of non-compact CYs, this has led to a complete solution of Vafa-Witten theory on rational surfaces. In the case of compact CYs, an approach to fix the ambiguity has been developed for a set of one-parameter threefolds. It relies on wall-crossing and solution of topological string by the direct integration method. I'll explain how the resulting generating functions allow to overcome the limitations of this method and find new Gromov-Witten invariants inaccessible by other means.