Graded rings and birational geometry

This paper is a written version of my lecture \Rings and varieties" at the Kinosaki algebraic geometry workshop in Oct 2000, and a series of two lectures at Tokyo University in Dec 2000. It is intended to be informative and attractive, rather than strictly accurate, and I expect it to stimulate work in a rapidly developing eld (as did its predecessor Reid [R3]). The paper was prepared in a hurry to meet a deadline, and one or two sections remain in rst draft. I apologise to the reader