The Cauchy problem for the quasilinear Schrodinger equatio

These are lectures notes for a 6 hour course given at PCMI, Park City, Utah, in the summer of 2003. The notes are based on a series of joint works of Kenig-Ponce-Vega. Each lecture had problems assigned with it. The lecture notes were prepared with the help of Justin Holmer. Justin Holmer also provided solutions to the problems, in an appendix to the notes.