Abstract: In this talk, I will provide a brief introduction to the area of K-stability. Roughly, K-stability is an algebro-geometric stability condition on varieties, whose motivations come from both algebraic and complex differential geometry. On the algebraic geometry side, K-stability provides the correct notion of stability for forming moduli spaces of Fano varieties. On the differential geometry side, K-stability is conjecturally equivalent to the existence of a Kähler metric of constant scalar curvature. It was in this latter context that K-stability first arose, in the work of Tian and Donaldson. After giving some motivation for why K-stability is important, I will describe the definition in basic terms, then finish by giving a few examples of how K-stability can be computed.