The theory of Bridgeland stability conditions assigns a complex manifold to the derived category of coherent sheaves on a smooth projective variety. The structure of this complex manifold is central for applications to algebraic geometry and mathematical physics, but describing even a connected component is often a difficult, open problem. In this talk, I will describe a conjectural framework for the stability manifolds of abelian varieties and their noncommutative analogues using generalized complex tori, extending a conjecture of Kontsevich. As applications, I will describe the stability manifold of any abelian threefold and complete the description of mirror symmetry for abelian varieties, after Golyshev-Lunts-Orlov. This is based on work in progress, joint with Fabian Haiden.