We will review various aspects of the self-avoiding walk model. This model, inspired by the physics of polymers, is a source of combinatorial and probabilistic problems that are of the best kind: they are simple to state but difficult to solve. In this talk, we propose to describe some of these problems. In particular, we will present the proof of a recent theorem, obtained jointly with Stanislav Smirnov, concerning the number of self-avoiding walks on the honeycomb lattice.