This talk with describe an existence theory and a computational method for traveling wave solution of the Euler quations for a fluid with a free surface. The approach uses Zakharov's Hamiltonian form of the equations of motion in an essential way, posing the equations of motion in surface variables in terms of the Dirichlet-Neumann operator for the fluid domain. Both two- and three-dimensional solutions are obtained. The existence theory is in the case in which at least some surface tension is present, and there is a close analogy with the resonant Lyapunov center theorem. The numerical computations of nonlinear and quite steep surface water waves are based on these surface variables, and use a surface spectral method and a Taylor expansion of the Dirichlet-Neumann operator which is an extension of the Hadamard variational formula for the Green's function. A clear distinction between shallow water and deep water waves is observed. This work is in collaboration with D. Nicholls.