Abstract: Kontsevich and Soibelman suggested a correspondence between Donaldson-Thomas invariants of Calabi-Yau 3-folds and holomorphic curves in complex integrable systems. After reviewing this general expectation, I will present a concrete example related to mirror symmetry for the local projective plane (partly joint work with Descombes, Le Floch, Pioline), along with applications in enumerative geometry (partly joint work with Fan, Guo, Wu). I will end by an “explanation” of the general correspondence based on the physics of 4d N=2 quantum field theories and holomorphic Floer theory.