abstract: An extremal Kahler metric is a kind of canonical metric on a complex manifold, which often exists when Kahler-Einstein metrics do not. In the search for such metrics one wants to know how they can degenerate, and in joint work with X. Chen, we show that under energy and non-collapsing assumptions degeneration can lead only to orbifold-type singularities. In the second half of the talk, we explain an application, from joint work of Chen-LeBrun-Weber, where a new Einstein manifold was discovered.