In this talk we investigate analyticity properties and High-Order Spectral numerical simulation of Dirichlet-Neumann operators which arise in boundary value and free boundary problems from a wide variety of applications (e.g., fluid and solid mechanics, electromagnetic and acoustic scattering). More specifically, we consider DNO defined on domains inspired by the simulation of ocean waves subject to quasi-periodic boundary conditions. Our theory shows that the DNO, when perturbed from a flat interface configuration, is parametrically analytic (as a function of deformation height/slope) for profiles of finite smoothness. The method of proof suggests a stable and highly accurate method of numerical simulation that we describe and verify with careful experiments.