Stokes waves, which are small amplitude space periodic solutions of the gravity water waves, are unstable to long-wave perturbations, as Benjamin and Feir predicted in1967. In this talk I will describe a series of recent results that describe the spectrum of the linearized operator at a Stokes wave, both near the origin – proving that such spectrum depicts a complete “figure 8” – and away from the origin – proving the existence of isolas of instability. This is based on a series of works with M. Berti, L. Corsi and P. Ventura.