We will discuss the basics of extremal Kahler metrics, following the paper in which Eugenio Calabi first introduced them. We will discuss their definition and some equivalent formulations. We will characterize extremal Kahler metrics in complex dimension 1, where they have constant scalar curvature. Every Riemann surface admits such a metric, and extremal metrics were introduced in order to help generalize this result to higher dimensional Kahler manifolds. We will describe examples of manifolds in higher dimensions which admit extremal metrics but cannot admit Kahler metrics of constant scalar curvature, and examples of manifolds which cannot admit extremal metrics at all.