  This manuscript explores the connections between a class of stochastic
processes called "Stochastic Loewner Evolution" (SLE) and conformal field
theory (CFT).
  First some important results are recalled which we utilise in the sequel, in
particular the notion of conformal restriction and of the "restrcition
martingale", originally introduced in Conformal restriction (G.F. Lawler, et
al).
  Then an explicit construction of a link between SLE and the representation
theory of the Virasoro algebra is given. In particular, we interpret the Ward
identities in terms of the restriction property and the central charge in terms
of the density of Brownian bubbles. We then show that this interpretation
permits to relate the $\kappa$ of the stochastic process with the central
charge $c$ of the conformal field theory. This is achieved by a highest-weight
representation which is degenerate at level two, of the Virasoro algebra.
  Then we proceed by giving a derivation of the same relations, but from the
theoretical physics point of view. In particular, we explore the relation
between SLE and the geometry of the underlying moduli spaces.
  Finally we outline a general construction which allows to construct random
curves on arbitrary Riemann surfaces. The key to this is to consider the
canonical operator $\frac{\kappa}{2}L^2_{-1}-2L_{-2}$ in conjunction with a
boundary field that is a degenerate highest-weight field $\psi$ as the
generator of a diffusion on an appropriate moduli space.
