Alexandrov geometry reflects the geometry of Riemannian manifolds when stripped from everything but their structure as metric spaces with a (local) lower curvature bound. In this talk I will define Alexandrov spaces and discuss basic properties, constructions and examples. By now there are numerous applications of Alexandrov geometry, including Perelman's spectacular solution of the geometrization conjecture for 3-manifolds.

The utility of Alexandrov geometry to Riemannian geometry is due to a large extend by the fact that there are several geometrically natural constructions that are closed in Alexandrov geometry but not in Riemannian geometry. These include, but are not limited to (1) Taking Gromov-Hausdorff limits, (2) Taking quotients, and (3) forming cones, jones etc of positively curved spaces. In the talk I will give examples of applications of each of these and one additional new construction.