We discuss some recent developments in the problem of Khler metrics of constant scalar curvature and stability in geometric invariant theory. In particular, we discuss various notions of stability, both finite and infinite-dimensional, and various analytic methods for the problem. These include estimates for energy functionals, density of states and Tian-Yau-Zelditch and Lu asymptotic expansions, geometric heat flows, and both a priori estimates and pluripotential theory for the complex Monge-Ampre equation.