  This paper describes joint work with Oded Schramm and Wendelin Werner
establishing the values of the planar Brownian intersection exponents from
which one derives the Hausdorff dimension of certain exceptional sets of planar
Brownian motion. In particular, we proof a conjecture of Mandelbrot that the
dimension of the frontier is 4/3. The proof uses a universality principle for
conformally invariant measures and a new process, the stochastic Loewner
evolution ($SLE$), introduced by Schramm. These ideas can be used to study
other planar lattice models from statistical physics at criticality. I discuss
applications to critical percolation on the triangular lattice, loop-erased
random walk, and self-avoiding walk.
