This talk will be an exposition of the relation between the integrable structure of the KdV equation and Kodaira-Titchmarsh spectral theory for Sturm-Liouville operators. Though this an old topic, the novelty here is an approach that applies to a very broad class of functions: basically any initial data that is bounded  below.   This will be used to explain a conjecture, due to McKean, for a very simple measure-theoretic characterization of the invariant manifolds of KdV. This has a natural extension to other classes of integrable, dispersive PDE such as Nonlinear Schrodinger. This is also the motivation for recent work with Dylan Murphy on discrete analogues; particularly, the Toda lattice hierarchy and the spectral theory of discrete Sturm-Liouville operators.