This thesis studies the problem of using surface splines as an approximation basis. The simplified surface spline basis, a slight simplification of the C1-surface spline construction of Joerg Peters, is presented. The simplified surface does not guarantee a C1 surface, but has explicit formulas, explicit basis functions and known dimension. In addition, it resides close to the C1-surface spline. The simplified surface spline basis is employed in interpolation, least squares approximation and faired least squares approximation. In addition, the mesh notation, a notation for writing algorithms on polyhedral meshes in a concise and mathematical way, is presented.