A lattice walk is said to be self-avoiding if it never visits the same vertex twice. These simple objects were introduced in physics in the
1940's as a natural model of polymers. Since then, their study, fuelled by beautiful predictions often coming from statistical
physics, has become an important question in combinatorics and probability theory.
As for many lattice models, the properties of self-avoiding walks (SAWs) are better understood in high dimension. Roughly speaking, in dimension 5 and beyond, the properties of SAWs resemble those of random walks (Hara-Slade, 1992). I will focus on the tricky dimension 2, where the most elementary questions remain unsolved: what is, asymptotically, the number of n-step SAWs? What is, on average, their end-to-end distance? Simple answers to these questions have been conjectured decades ago, yet they have resisted all proving attempts so far.
I will describe some classical tools, like unfolding and pivot moves. I will also cover a recent major progress due to Duminil-Copin and Smirnov, which deals with SAW on the hexagonal lattice, and some variations on this result.