There are recent lecture notes by Bauerschmidt, Duminil-Copin, Goodman, and Slade (arXiv:1206.2092). Most of important papers should be in the references to that lecture notes, for example:
G.F. Lawler, O. Schramm, and W. Werner, On the scaling limit of planar self-avoiding walk.
H. Duminil-Copin and S. Smirnov, The connective constant of the hexagonal lattice equals $\sqrt{2+\sqrt2}$.