The intermediate long wave equation (ILW) is a model for internal waves in a stratified fluid. The ILW is not just a completely integrable PDE - it also provides a bridge between two other integrable PDEs: the Korteweg-de Vries and Benjamin-Ono equations. In this talk, I will present some recent results on the ILW. In particular, I will discuss a priori estimates in Sobolev spaces and the regularity gain of the solution map in the shallow water limit. This is joint work with Rowan Killip and Monica Vișan and with Andreia Chapouto, Guopeng Li, and Tadahiro Oh.