  SLE is a random growth process based on Loewner's equation with driving
parameter a one-dimensional Brownian motion running with speed $\kappa$. This
process is intimately connected with scaling limits of percolation clusters and
with the outer boundary of Brownian motion, and is conjectured to correspond to
scaling limits of several other discrete processes in two dimensions.
  The present paper attempts a first systematic study of SLE. It is proved that
for all $\kappa\ne 8$ the SLE trace is a path; for $\kappa\in[0,4]$ it is a
simple path; for $\kappa\in(4,8)$ it is a self-intersecting path; and for
$\kappa>8$ it is space-filling.
  It is also shown that the Hausdorff dimension of the SLE trace is a.s. at
most $1+\kappa/8$ and that the expected number of disks of size $\eps$ needed
to cover it inside a bounded set is at least $\eps^{-(1+\kappa/8)+o(1)}$ for
$\kappa\in[0,8)$ along some sequence $\eps\to 0$. Similarly, for $\kappa\ge 4$,
the Hausdorff dimension of the outer boundary of the SLE hull is a.s. at most
$1+2/\kappa$, and the expected number of disks of radius $\eps$ needed to cover
it is at least $\eps^{-(1+2/\kappa)+o(1)}$ for a sequence $\eps\to 0$.
