We consider the scattering from and transmission through a one-dimensional periodic surface. For this problem, the electromagnetic cases of TE and TMpolarization reduce to the scalar acoustic examples. Three different theoretical and computational methods are described, all involving the solution of integral equations and their resulting discrete matrix system of equations for the boundary unknowns. They are characterized by two sample spaces for their discrete solution, coordinate (C) space and spectral (S) space, and labelled by the sampling of the rows and columns of the discretized matrices. They are coordinate-coordinate (CC), the usual coordinate-space method, spectral-coordinate (SC) where the matrix rows are discretized or sampled in spectral space, and spectral-spectral (SS) where both rows and columns are sampled in spectral space. The SS method uses a new topological basis expansion for the boundary unknowns. Equations are derived for infinite surfaces, then specialized and solved for periodic surfaces. Computational results are presented for the transmission problem as a function of roughness, near grazing incidence as well as many other angles, density and wavenumber ratios. Matrix condition numbers and different sampling method are considered. An error criterion is used to gauge the validity of the results. The computational results indicated that the SC method was by far the fastest (by several orders of magnitude), but that it became ill-conditioned for very rough surfaces. The CC method was most reliable, but often required very large matrices and was consequently extremely slow. It is shown that the SS method is computationally efficient and accurate at near grazing incidence and can be used to fill a gap in the literature. Extensive computational results indicate that both SC and SS are highly robust computational methods. Spectral based methods thus provide viable computational schemes to study periodic surface scattering.