After highlighting the importance of K-stability for algebraic geometers with some historic remarks, I will survey some techniques that have been developed to determine K-stability of Fano varieties. The latter will be illustrated by examples, focusing on the practical and powerful Abban-Zhuang method. I will then show how this approach can be interpreted in convex geometry perspective, with some new insights. As an application, I will show that every smooth index one hypersurface in a quadric is K-stable.