Recent developments on the well-posedness of dispersive equations

We will review the progress in the last 20 years on the well-posedness theory of dispersive equations, concentrating in the case of the Korteweg- de Vries equation to illustrate it. We will then describe some of the current challenges in the area, and biefly discuss recent joint works ( mainly with Alex Ionescu ) in which progress has been made for the Benjamin-Ono equation, the Schrodinger map system and the Kadomstev-Petviashvili I equation.