Explore the powerful applications of Ricci flow in deforming and regularizing manifolds in this 47-minute lecture by Peter Topping from the University of Warwick, presented at the Institut des Hautes Etudes Scientifiques (IHES). Discover how Ricci flow can transform manifolds with specific geometric or topological conditions into identifiable special manifolds, such as deforming positively curved manifolds into spherical space forms. Delve into recent applications of Ricci flow's regularization effect, including proving that certain metric limit spaces are manifolds, regularizing manifold geometry at infinity for pinching theorems, and identifying new expanding Ricci solitons. Gain insights into selected examples that demonstrate the versatility and power of Ricci flow in modern geometric analysis.