This paper discusses the bifuraction theory for the equations for traveling surface water waves, based on the formulation of Zakharov and of Craig and Sulem in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions, the phase velocity is a two dimensional vector, and the resulting bifurcation equations describe two dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet-Neumenn operator for the fluid domain. This lends itself naturally to numerical computation through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls. We present a perturbation analysis of the resulting bifurcation surfaces for the three dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three dimensional shallow water regime, and their transition to high aspect ratio rectangular patters as the depth increases to infinity.