Geometric domain decomposition methods are described for solving x-y geometry discrete ordinates (SN) problems on parallel architecture computers. First, a parallel source iteration scheme is developed; here, one subdivides the spatial domain of the problem, performs transport sweeps independently in each subdomain, and iterates on the scattering source and the interface fluxes between each subdomain. Second, a parallel diffusion synthetic acceleration (DSA) scheme is developed to speed up the convergence of the parallel source iteration. These schemes have been implemented on the IBM RP3, a shared/distributed memory parallel computer. The numerical results show that the parallel source iteration and DSA methods both exhibit significant speedups over their scalar counterparts, but that a degradation in parallel efficiency occurs due to the geometric domain decomposition (iteration on interface fluxes) and the overhead time required for the communication of data between processors. However, the degradation due to geometric domain decomposition is unimportant if the subdomains are not optically thin or do not contain a small number of cells.