Explore a comprehensive lecture on the K-stability of Fano varieties, delivered by Chenyang Xu from Princeton University. Delve into this central topic in complex geometry and its relation to the Kahler-Einstein problem. Discover how the machinery of higher dimensional geometry, particularly the minimal model program, serves as a fundamental tool in studying K-stability. Learn about the significant progress made in solving major conjectures, including the Yau-Tian-Donaldson Conjecture and the construction of a moduli space for Fano varieties. Gain insights into recent developments in this field, covering topics such as variational viewpoint, K-polystability, origins, theorems, basis type dividers, K-module space, good module space, hypersurface, and projectivity. This 58-minute talk, part of a joint presentation with Ziquan Zhuang, offers a thorough survey of the subject and sets the stage for a more detailed discussion in the subsequent part.