A fundamental theme in Kahler geometry in the last 30 years has been the connection between canonical geometric objects, usually described by nonlinear PDEs, and algebro-geometric stability. I will discuss two concrete examples of this-- the J-equation which arises in the study of constant scalar curvature Kahler metrics, and the Lagrangian phase equation, which arises from mirror symmetry. In each of these cases the relevant stability conditions are simple to write down, and play a direct role in proving estimates for the corresponding nonlinear equation. This talk will discuss joint work with G. Szekelyhidi, A. Jacob, and S.-T. Yau.