Recent progress in 2D statistical physics

Two-dimensional models of statistical physics have long been studied by physicists, using tools such as quantum and conformal field theories and renormalization groups as well as through explicit computations in integrable cases. On the mathematics front, two objects were introduced over the last decade, shedding new light on their geometry: first, stochastic Loewner evolutions, proved by Schramm to be the unique possible scaling limits of models exhibiting conformal invariance; second, (para)fermionic observables, used by Smirnov to actually prove conformal invariance of several of them. I will present a panorama of these recent advances and some of the most puzzling open questions remaining to be solved.