Explore the profound implications of Thurston's geometric perspective on 3-manifolds and Perelman's Geometrization Theorem in this advanced mathematics lecture. Delve into the fundamental role of hyperbolic geometry in addressing topological questions in three dimensions. Examine algorithmic and group-theoretic outcomes of Geometrization, including solutions to the homeomorphism problem and residual finiteness of fundamental groups in compact 3-manifolds. Witness practical demonstrations of discovering hyperbolic structures on 3-manifolds and their application in extensive computational experiments. This 55-minute talk, part of the Workshop on Groups around 3-Manifolds, offers a deep dive into cutting-edge research in 3-manifold topology and its far-reaching consequences.