We consider problems of smooth nonconvex optimization: unconstrained, bound-constrained, and with general equality constraints. We show that algorithms for these problems that are widely used in practice can be modified slightly in ways that guarantees convergence to approximate first- and second-order optimal points with complexity guarantees that depend on the desired accuracy. The methods we discuss are constructed from Newton's method, the conjugate gradient method, log-barrier method, and augmented Lagrangians. (In some cases, special structure of the objective function makes for only a weak dependence on the accuracy parameter.) Our methods require Hessian information only in the form of Hessian-vector products, so do not require the Hessian to be evaluated and stored explicitly. This talk describes joint work with Clement Royer, Yue Xie, and Michael O'Neill.