While initially introduced as a model for water waves, the
Korteweg -- de Vries equation has grown to be one of the most-studied
partial differential equations.  In this talk, I will present a proof of
well-posedness for KdV for initial data which is merely in H^{-1}.
The argument applies equally well for KdV posed on the line or the
circle.  This is based on joint work with Rowan Killip.