We present a Hamiltonian formulation for three-dimensional surface water waves based on potential flow theory. In this formulation, the problem is reduced to a lower-dimensional one involving boundary variables alone. This is accomplished by introducing the Dirichlet-Neumann operator which expresses the normal fluid velocity at the free surface in terms of the velocity potential there, and in terms of the surface variations which determine the fluid domain. A Taylor series expansion of the Dirichlet- Neumann operator in homogeneous powers of the surface variations is proposed. This formulation has implications for the convenience of perturbation calculations and numerical simulations of the full Euler equations for water waves. An efficient and accurate spectral method based on the fast Fourier transform is developed to evaluate numerically the Dirichlet-Neumann operator and solve the full Euler equations. Some numerical applications will be presented.