I will consider classical 2D traveling water waves with vorticity. By
means of local and global bifurcation theory using topological degree,
we can prove that there exist many such waves. They are exact smooth
solutions of the Euler equations with the physical boundary conditions.
Some of the waves are quite tall and steep and some are overhanging.
There are periodic ones and solitary ones. I will exhibit some
numerical computations of such waves. New analytical results will be
presented on waves with favorable vorticity.