  We consider the two-dimensional problem for steady water waves with vorticity
on water of finite depth. While neglecting the effects of surface tension we
construct connected families of large amplitude periodic waves approaching the
limiting wave, which is either a solitary wave, the highest solitary wave, the
highest Stokes wave or a Stokes wave with a breaking profile. In particular,
when the vorticity is nonnegative we prove the existence of highest Stokes
waves with an included angle of 120 degrees. In contrast to previous studies we
fix the Bernoulli constant and consider the wavelength as a bifurcation
parameter, which guarantees that the limiting wave has a finite depth. In fact,
this is the first rigorous proof of the existence of extreme Stokes waves with
vorticity on water of finite depth. Beside the existence of highest waves we
provide a new result about the regularity of Stokes waves of arbitrary
amplitude (including extreme waves). Furthermore, we prove several new facts
about steady waves, such as a lower bound for the wavelength of Stokes waves,
while also eliminate a possibility of the wave breaking for waves with
non-negative vorticity.
