From Stationary Phase to Steepest Descent

Our aim here is to clarify the distinction between the nonlinear-stationary-phase idea and the nonlinear-steepest-descent idea, stressing the importance of actual steepest-descent contours in some problems. We mostly use the nonlinear Schr\"odinger equation as our working model, but we also digress to the KdV equation at some point. This is a slightly revised version of a review paper that will appear in the forthcoming volume (Contemporary Mathematics, AMS) honoring Percy Deift.