We will discuss the algebraic K-stability theory of Fano varieties, including the following topics: the foundational theory of K-stability, including the original definition of K-stability, the minimal model program of family of Fano varieties, etc.; the construction of K-moduli spaces, including the openness of K-semistability, the uniqueness of S-equivalence for K-semistable degeneration, and the existence of good moduli for K-polystable Fano varieties; and, verifying explicit examples of Fano varieties to be K-stable, including cubic threefolds, smooth hypersurfaces low Fano index, etc.