I will introduce the manifold of Bridgeland stability conditions on a triangulated category C and explain some geometric motivations for its study. I will explain how certain divergent paths in this manifold correspond to categorical structures on C. Furthermore, I will illustrate a modular partial compactification of (a quotient of) the stability manifold in which these paths converge. This is based on joint work with Daniel Halpern-Leistner and Jeffrey Jiang.