Fano varieties are complex algebraic varieties admitting positive Ricci curvature metrics. They form one of the three fundamental building blocks of algebraic varieties, thus their classification problem is important. However, it is known that moduli spaces of all Fano varieties have pathological behaviors. In this talk, I will explain that if we impose K-stability on Fano varieties, an algebraic condition arising from the study of K\"ahler-Einstein metrics, then we indeed get a compact moduli space. Based on joint works with H. Blum, D. Halpern-Leistner, C. Xu, and Z. Zhuang. Speaker(s): Yuchen Liu (Northwestern University)