Conformal field theory has been very successful in describing the low-energy properties of 2D statistical systems at their critical points.  Among the most beautiful results in that field is the description of geometrical properties at criticality. A famous example of this is the exact determination of the fractal dimension of magnetisation domains in the Ising model.  Beyond the field theory toolbox, a mathematically rigorous description of such objects has been introduced in 2000: the Schramm-Loewner Evolution (SLE).
In my talk I will give a simple introduction to these concepts in the context of loop models, which are a 2D version of the spin O(n) model in higher dimension.  I will focus on the surface critical behaviour of these models and the link with the SLE approach.  After that I will quickly present some recent results about the domain-walls in the Potts model, which are no longer loops but critically branching curves.