Stokes in his classical memoir made many contributions about periodic traveling waves at the free surface of a water flow. In particular, he conjectured that a wave of greatest possible height would exhibit stagnation with a 120 degree's corner. In the zero vorticity case, Amick, Fraenkel, and Toland answered the conjecture affirmatively. But interior stagnation is not allowed for all waves.

The situation becomes much more complicated with non-zero vorticity. Recently, Constantin, Strauss, and Varvaruca worked out global bifurcation in the constant vorticity case, and they conjectured about limiting waves. I will present the joint work with Sergey Dyachenko (Illinois), numerically computing Stokes waves with constant vorticity from zero close to the limiting wave, and discovering new limiting waves.