Kuznetsov showed the derived category of a cubic fourfold contains a special subcategory which can be thought of as a "noncommutative K3 surface". The goal of this talk is to describe the structure of moduli spaces of Bridgeland stable objects on this noncommuative K3 surface, together with several applications. This is joint work with Bayer, Lahoz, Macri, Nuer, and Stellari.