The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. Closely related to the KdV and the Benjamin-Ono equation, the ILW equation is formally completely integrable. The integrability properties of KdV and BO also have connections to their sharp well-posedness theory via Jost-Pais type theorems. In this talk, I will give an introduction to the inverse scattering method for KdV, and present some recent results on the ILW direct scattering problem, together with a Jost-Pais theorem for it.