The streamlines of periodic irrotationaltraveling water waves are known to be real-analytic,with exception of the free surface in the case thewave of greatest height which has a corner at the wavecrest (the lateral tangents being at an angle of 2pi/3).The regularity of waves of small and moderate amplitudeis, perhaps surprisingly, little affected by thepresence of vorticity in the flow. This is joint workwith J. Escher.