**Abstract:** After setting the stage by sketching a few facts about the topology and geometry of surfaces, I will explain why the study of the topology of 3-dimensional manifolds is inextricably linked to the study of homogenous geometries such as Euclidean, spherical, and (especially) hyperbolic geometry. This perspective, introduced by Thurston in the 1980s, was stunningly confirmed in the early 2000s by Perelman's deep work using geometric PDEs, and lead to the solution of the 100 year-old Poincaré conjecture. I will hint at how this perspective brings other areas of mathematics, specifically algebraic geometry and number theory, to bear on problems that initially appear purely topological in nature, and conclude with a live computer demonstration of how geometry can be used to tell different 3-manifolds apart in practice.