  The aim of these notes is threefold. First, we discuss geometrical aspects of
conformal covariance in stochastic Schramm-Loewner evolutions (SLEs). This
leads us to introduce new ``dipolar'' SLEs, besides the known chordal, radial
or annular SLEs. Second, we review the main features of our approach connecting
SLEs to conformal field theories (CFTs). It is based on using boundary CFTs to
probe the SLE hulls. Finally, we study zig-zag probabilities and their relation
with CFT correlation functions. We suggest a putative link between the braiding
of SLE samples and that of CFT correlation functions.
