Appears in collection : Mathematics on the Crossroad of Centuries - A Conference in Honor of Maxim Kontsevich's 60th Birthday

Our aim is to use the DT invariants to of a CY3 triangulated category to define a geometric structure on the space of stability conditions. So far we only know how to do this in a few simple examples. In the talk I will describe the expected geometry, which involves a hyperkahler structure, and discuss sthe main class of examples, which are some kind of complexified Hitchin systems.