On geometric properties of the Schramm-Loewner evolution

The Schramm-Loewner evolution (SLE) is a family of random fractal curves that arise in a natural manner as scaling limits of interfaces in planar critical lattice models from statistical mechanics. The SLE curves exhibit many interesting geometric structures also present in the related models, now rigorously accessible through a combination of complex analytic and probabilistic techniques. I will give an overview of some known properties and open questions regarding the fine geometry of SLE curves. The talk is in part based on joint works with G. Lawler, with S. Rohde and C. Wong, and with T. Alberts and I. Binder.