A number of lattice models in two-dimensional statistical physics are conjectured to exhibit conformal invariance in the scaling limit at criticality. In this talk, I will try to explain what the previous sentence means, focusing on three elementary examples: simple random walk, self-avoiding walk, loop-erased random walk, percolation, Ising model I will describe the limit objects, Schramm-Loewner Evolution (SLE), the Brownian loop measure, and the normalized partition functions, and show how conformal invariance can be used to calculate quantities ("critical exponents") for the model. I will also describe why (in some sense) there is only a one-parameter family of conformally invariant limits. In conformal field theory, this family is parametrized by central charge.

Much of the talk will be based on joint work with Oded Schramm and Wendelin Werner although I will mention work by a number of different researchers.