Associated to a K¨ahler form ω on a compact K¨ahlerian manifold N is its Calabi energy— the L 2 -norm of its scalar curvature, computed with respect to the natural volume form. It is of geometric interest to find K¨ahler metrics that are critical for the Calabi energy among representatives of a fixed de Rham class. Critical metrics, which are necessarily minima, are called extremal K¨ahler metrics. In this talk, I will survey known facts about extremal metrics, then describe how the momentum function associated to a circle action can be used to construct extremal metrics on certain manifolds obtained by compactifying suitable holomorphic line bundles. There is a necessary and sufficient complex-analytic criterion for these spaces to admit metrics of constant scalar curvature. This condition can be phrased in terms of a moment polytope, which yields nice insights into the geometry of these metrics.