  In these Notes, a comprehensive description of the universal fractal geometry
of conformally-invariant scaling curves or interfaces, in the plane or
half-plane, is given. The present approach focuses on deriving critical
exponents associated with interacting random paths, by exploiting their
underlying quantum gravity structure. The latter relates exponents in the plane
to those on a random lattice, i.e., in a fluctuating metric, using the
so-called Knizhnik, Polyakov and Zamolodchikov (KPZ) map. This is accomplished
within the framework of random matrix theory and conformal field theory, with
applications to geometrical critical models, like Brownian paths, self-avoiding
walks, percolation, and more generally, the O(N) or Q-state Potts models and,
last but not least, Schramm's Stochastic Loewner Evolution (SLE_kappa).
  These Notes can be considered as complementary to those by Wendelin Werner
(2006 Fields Medalist!), ``Some Recent Aspects of Random Conformally Invariant
Systems,'' arXiv:math.PR/0511268.
