A self-avoiding walk (SAW) on a graph is a path which does not visit any vertex twice. In this talk, we study an enumeration problem consisting in counting such walks of given lengths. More precisely, we will present the proof (obtained jointly with S. Smirnov) of a conjecture of Nienhuis stating that the number of SAWs of length $n$ on the hexagonal lattice grows like $\sqrt{2+\sqrt 2}^{n+o(n)}$. The proof will also shed new light on a very instructive and beautiful phase transition in the geometric properties of long SAWs.