If the sectional curvature of a manifold is bounded below by a constant K, then geodesics emanating from a point diverge no fasted than geodesics making the same angle in the simply connected surface of constant curvature K. This fundamental observation, due to Toponogov, leads to the theory of Alexandrov spaces, which are metric spaces in which curvature is bounded below by a constant K in the Toponogov sense. These spaces have much structure, leading to a detailed understanding of Alexandrov spaces of dimension at most 2. These in turn of limits of sequences of 3-manifolds with collapsing volume, allowing for a complete topological understanding of the latter. This understanding is a crucial ingredient in Perelman's proof of the Geometrization Conjecture for 3-manifolds.