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In this talk, we will discuss the existence problem of extremal metrics on a Kähler manifold. The best known examples of these are Kähler-Einstein and constant scalar curvature Kähler (cscK) metrics. Yau's resolution of the Calabi conjecture proves existence of Kähler-Einstein metrics on manifolds with vanishing first Chern class. Recently, through the work of Chen-Cheng, cscK metrics can be shown to exist under certain stability conditions. The focus of this talk will be an overview of the analytic methods used in both of these proofs.