Spaces with curvature bounded below, or Alexandrov spaces, are defined as complete metric length spaces satisfying, locally, certain distance comparison condition. For Riemannian metrics this condition is equivalent to the lower bound on sectional curvatures. In fact, this local condition implies the similar comparison in the large -- for Riemannian metrics this is the celebrated Toponogov comparison theorem. Therefore, the class of Alexandrov spaces with fixed lower curvature bound is closed w.r.t. Gromov-Hausdorff convergence, in particular, it contains all the limits of complete Riemannian metrics with sectional curvatures uniformly bounded below.
Alexandrov spaces may have metric singularities (like those of the intrinsic metric of a convex hypersurface), as well as topological ones (like those of the quotient of a Riemannian manifold by an isometric group action).
However, finite-dimensional Alexandrov spaces have a number of nice properties:
There exists a unique tangent cone at each point which is itself a non-negatively curved Alexandrov space of the same dimension.
The singular set (i.e. the set of points where tangent cone is not isometric to Euclidean space or half-space) has Hausdorff codimension at least 2 (it may be dense, though).
A small open spherical neighborhood of each point is homeomorphic to the tangent cone at this point.
An Alexandrov space has a canonical stratification. The strata are totally geodesic in a certain sense (but not necessarily Alexandrov spaces themselves). Each stratum is a topological manifold.
Other results and open problems will be discussed.