  We review some of the results that have been derived in the last years on
conformal invariance, scaling limits and properties of some two-dimensional
random curves. In particular, we describe the intuitive ideas that lead to the
definition of the Schramm-Loewner evolutions SLE, we define these objects,
study its various properties, show how to compute (probabilities, critical
exponents) using SLE, relate SLE to planar Brownian motions (i.e. the
determination of the critical exponents), planar self-avoiding walks, critical
percolation, loop-erased random walks and uniform spanning trees.
