The search for canonical metrics, often a minimizer of a normalized quadratic curvature functional, is a central problem in differential geometry. Attempts to find such metrics directly have proven difficult, in part due to collapsing phenomena which make some analytic tools inapplicable. Indirect approaches have been successful in special cases, in one case leading to the discovery of an Einstein metric on CP^2 # 2 CP^2 . In this talk we will study approaches to understanding the Gromov-Hausdorff compactification of the moduli space of extremal Kaehler metrics, based on work of Chen-Weber, Chen-LeBrun- Weber, and Weber. We will describe new analytic-topological and complex analytic methods, special to extremal Kaehler metrics in dimension 4, that can be used to obtain regularity and convergence results.