2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119824In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.This is a revised version of the article by the same authors that originally appeared in Asian J. Math., {\bf 10}({\bf 2}) (2006), 165--492Differential Geometry53C21, 53C44text