Explore the profound implications of Thurston's geometric perspective on 3-manifolds and Perelman's Geometrization Theorem in this 59-minute lecture by Nathan Dunfield. Delve into the fundamental concepts of 3-manifold topology before examining how hyperbolic geometry plays a crucial role in addressing topological questions in three dimensions. Discover the algorithmic and group-theoretic consequences of Geometrization, including solutions to the homeomorphism problem and residual finiteness of fundamental groups in compact 3-manifolds. Gain practical insights into finding hyperbolic structures on 3-manifolds and their application in large-scale computational experiments. This talk, part of the Workshop on Groups around 3-Manifolds at the Centre de recherches mathématiques, offers a comprehensive exploration of advanced topics in geometric topology.