  We discuss the possible candidates for conformally invariant random
non-self-crossing curves which begin and end on the boundary of a multiply
connected planar domain, and which satisfy a Markovian-type property. We
consider both, the case when the curve connects a boundary component to itself
(chordal), and the case when the curve connects two different boundary
components (bilateral). We establish appropriate extensions of Loewner's
equation to multiply connected domains for the two cases. We show that a curve
in the domain induces a motion on the boundary and that this motion is enough
to first recover the motion of the moduli of the domain and then, second, the
curve in the interior. For random curves in the interior we show that the
induced random motion on the boundary is not Markov if the domain is multiply
connected, but that the random motion on the boundary together with the random
motion of the moduli forms a Markov process. In the chordal case, we show that
this Markov process satisfies Brownian scaling and discuss how this limits the
possible conformally invariant random non-self-crossing curves. We show that
the possible candidates are labeled by a real constant and a function
homogeneous of degree minus one which describes the interaction of the random
curve with the boundary. We show that the random curve has the locality
property if the interaction term vanishes and the real parameter equals six.
