We discuss the algebraic K-stability theory of Fano varieties, specifically 1) the foundational theory of K-stability, including the original definition of K-stability, the valuative criterion of K-stability, etc., 2) the construction of general K-moduli stacks/spaces, including various necessary ingredients, and 3) verifying explicit examples of Fano varieties to be K-stable, including explicit K-moduli construction and estimate of stability of thresholds.