Sporadic informal observations over several decades (and most recently in Lewis-Overton, 2013) suggest that quasi-Newton methods for smooth optimization can also work surprisingly well on nonsmooth functions. This talk explores this phenomenon from several perspectives. First, we show how Powell's original 1976 BFGS convergence proof for smooth convex functions in fact extends to some nonsmooth settings. Secondly, we study how repeated BFGS updating at a single fixed point can serve as a separation oracle (for the subdifferential).