  Conformally-invariant curves that appear at critical points in
two-dimensional statistical mechanics systems, and their fractal geometry have
received a lot of attention in recent years. On the one hand, Schramm has
invented a new rigorous as well as practical calculational approach to critical
curves, based on a beautiful unification of conformal maps and stochastic
processes, and by now known as Schramm-Loewner evolution (SLE). On the other
hand, Duplantier has applied boundary quantum gravity methods to calculate
exact multifractal exponents associated with critical curves.
  In the first part of this paper I provide a pedagogical introduction to SLE.
I present mathematical facts from the theory of conformal maps and stochastic
processes related to SLE. Then I review basic properties of SLE and provide
practical derivation of various interesting quantities related to critical
curves, including fractal dimensions and crossing probabilities.
  The second part of the paper is devoted to a way of describing critical
curves using boundary conformal field theory (CFT) in the so-called Coulomb gas
formalism. This description provides an alternative (to quantum gravity) way of
obtaining the multifractal spectrum of critical curves using only traditional
methods of CFT based on free bosonic fields.
