The mathematical study of water waves began with the derivation of the basic mathematical equations of fluids by Euler in 1752. Later, water waves played a central role in the work of Poisson, Cauchy, Stokes, Levi-Civita and many others. It remains a very active area to the present day.

I will consider a classical 2D traveling water wave with vorticity. By means of local and global bifurcation theory using topological degree, we now know that there exist many such waves (exact solutions of the Euler equations with the physical boundary conditions) of large amplitude. This work is joint with Adrian Constantin. I will also exhibit some numerical computations of such waves, obtained jointly with Joy Ko. Many fundamental problems remain open.