From both geometric and algebraic viewpoints, the self-correcting properties of BFGS updating are discussed. These properties, expressed in terms of the sequence of BFGS Hessian approximations, lead to useful properties of the corresponding inverse Hessian approximations. These latter properties are exploited in two proposed algorithms, one for stochastic (nonconvex) optimization and one for deterministic convex (nonsmooth) optimization. This talk describes these algorithms, discusses their convergence properties, and illustrates their practical performance.