In this talk, we present high order numerical methods for solving multiphysics problems. First, we investigate the coupling of surface flow with subsurface flow. Surface flow is characterized either by Stokes or Navier-Stokes equations whereas subsurface flow is characterized by Darcy equations. Special interface conditions are considered between the subregions. Locally conservative methods, such as discontinuous Galerkin or mixed finite element methods, are used. Optimal error estimates are obtained. Second, we consider both implicit and explicit discontinuous formulations of the incompressible two-phase flow problem. In particular, we show numerical convergence of the p-version. An interesting fact is that no slope limiters are needed for the implicit method.