This paper discusses the entropy formula for the Ricci flow and its geometric applications. Grisha Perelman introduces the Ricci flow equation, which describes the evolution of a Riemannian metric on a manifold. Hamilton proved that this equation has a unique solution for a short time for an arbitrary smooth metric on a closed manifold. The Ricci flow preserves the positivity of the Ricci tensor in dimension three and the curvature operator in all dimensions. Hamilton also proved that the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are pinched point-wise as the curvature becomes large. These results led to convergence theorems for the Ricci flow. Perelman discusses the long-term behavior of the Ricci flow and the challenges in analyzing ancient solutions. He also introduces the concept of surgery, where a cylindrical neck is cut open and small caps are glued to each boundary. Hamilton proposed that this procedure could be used to make the Ricci flow nonsingular, with normalized curvatures staying bounded. Perelman confirms this conjecture and shows that the region where this does not hold is locally collapsed with curvature bounded below. Perelman also discusses the application of the Ricci flow to the Hamilton-Tian conjecture concerning Kähler-Ricci flow on Kähler manifolds with positive first Chern class. He also explores the connection between the Ricci flow and the renormalization group flow in quantum field theory, suggesting that the Ricci flow must be gradient-like. Perelman presents a detailed analysis of the Ricci flow, including the monotonicity formula, the no breathers theorem, and the no local collapsing theorem. He also discusses the statistical analogy of the Ricci flow and its relation to entropy. The paper concludes with a sketch of the proof of the geometrization conjecture for closed three-manifolds.This paper discusses the entropy formula for the Ricci flow and its geometric applications. Grisha Perelman introduces the Ricci flow equation, which describes the evolution of a Riemannian metric on a manifold. Hamilton proved that this equation has a unique solution for a short time for an arbitrary smooth metric on a closed manifold. The Ricci flow preserves the positivity of the Ricci tensor in dimension three and the curvature operator in all dimensions. Hamilton also proved that the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are pinched point-wise as the curvature becomes large. These results led to convergence theorems for the Ricci flow. Perelman discusses the long-term behavior of the Ricci flow and the challenges in analyzing ancient solutions. He also introduces the concept of surgery, where a cylindrical neck is cut open and small caps are glued to each boundary. Hamilton proposed that this procedure could be used to make the Ricci flow nonsingular, with normalized curvatures staying bounded. Perelman confirms this conjecture and shows that the region where this does not hold is locally collapsed with curvature bounded below. Perelman also discusses the application of the Ricci flow to the Hamilton-Tian conjecture concerning Kähler-Ricci flow on Kähler manifolds with positive first Chern class. He also explores the connection between the Ricci flow and the renormalization group flow in quantum field theory, suggesting that the Ricci flow must be gradient-like. Perelman presents a detailed analysis of the Ricci flow, including the monotonicity formula, the no breathers theorem, and the no local collapsing theorem. He also discusses the statistical analogy of the Ricci flow and its relation to entropy. The paper concludes with a sketch of the proof of the geometrization conjecture for closed three-manifolds.