  We show that there exists a unique (up to multiplication by constants) and
natural measure on simple loops in the plane and on each Riemann surface, such
that the measure is conformally invariant and also invariant under restriction
(i.e. the measure on a Riemann surface S' that is contained in another Riemann
surface S, is just the measure on S restricted to those loops that stay in S').
We study some of its properties and consequences concerning outer boundaries of
critical percolation clusters and Brownian loops.
