Conformal restriction and related questions

This paper is based on mini-courses given in July 2003. Its goal is to give a self-contained sketchy and heuristic survey of the recent results concerning conformal restriction, that were initiated in our joint work with Greg Lawler and Oded Schramm, and further investigated in the last year in joint work with Roland Friedrich, Greg Lawler, and by Julien Dubedat. These notes can be viewed as complementary to my Saint-Flour notes.