Abstract. In recent work the author has explored the idea that the DT invariants of a three-dimensional Calabi-Yau category can be used to define a geometric structure on the space of stability conditions. The relevant structure was christened a Joyce structure and can be viewed as a kind of non-linear Frobenius structure. There are close links with hyperkahler structures and the work of Gaiotto, Moore and Neitzke, and also with recent work in physics on non-perturbative completions of partition functions. The main examples of Joyce structures considered so far involve moduli spaces of Higgs bundles and flat connections on Riemann surfaces, and are a kind of complexification of the Hitchin system. The rough plan for the three talks is (1) Introduction, and definition of Joyce structures, (2) Moduli-theoretic construction of Joyce structures on spaces of quadratic differentials, (3) Tau function associated to a Joyce structure.