I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow : $X_t=X_s\times X_{ss}$ which preserve the length parametrization. Above $X(s,t)$ is a curve in ${ℝ}^3$, $s\in ℝ$ the arclength parameter, and $t$ denote the temporal variable. This flow appeared for the first time in the work of Da Rios (1906) as a crude approximation to the evolution of a vortex filament under Euler equation, and it is intimately related to the focusing cubic nonlinear Schrödinger...