The self-avoiding walk is a simply stated model for polymer chains that is mathematically very challenging to analyze. In two dimensions, it is conjectured that the scaling limit of this process is conformally invariant. It is now known that assumption of conformal invariance uniquely specifies what the limiting process must be. It is a particular case of the Schramm-Loewner evolution (SLE). It is also closely related to other conformally invariant processes, e.g., Brownian motion (in particular, the outer boundary of a planar Brownian motion path) and the boundaries of percolation clusters.

My talk will introduce the problem of the self-avoiding walk, discuss work that I did in collaboration with Oded Schramm and Wendelin Werner, and then will discuss more recent work by a number of researchers.