In this thesis we consider two dimensional, half-plane, unsteady water wave problems and their solution by boundary integral methods. Well-posed boundary integral solutions and convergent numerical schemes exist within the literature under the restrictive assumption of periodicity in both the boundary and the boundary data (overturning, or breaking, waves are included). The boundary integral formulation presented within gives a well-posed solution and leads to a convergent numerical scheme without the restriction of periodicity in the boundary and boundary data (while still allowing overturning waves).