Abstract: We present some recent results on singular solutions of the problem of travelling gravity surface water waves on flows with vorticity. It has been known since the work of Constantin and Strauss in 2004 that there exist spatially periodic waves of large amplitude for any vorticity distribution. We show that, for any nonpositive vorticity distribution and for any value of the period of the waves, a sequence of large-amplitude regular waves converges in a weak sense to an extreme wave with stagnation points at its crests. A similar results holds for nonnegative vorticity distributions, under some smallness assumptions, and provided that the period of the waves is sufficiently large. The proof is based on new a priori estimates, obtained by means of the maximum principle, for the fluid velocity and the wave height along the family of regular waves whose existence was proved by Constantin and Strauss. We also show, by new geometric methods, that this extreme wave has corners of 120 degrees at its crests, as conjectured by Stokes in 1880. The results presented were obtained in collaboration with Ovidiu Savin and Georg Weiss.