Currently, the problem size that can be solved in reasonable time using the Method of Moments (MoM) is limited by the amount of memory installed on the computer. This paper offers a new development which not only breaks this memory constraint, but also maintains the efficiency of running the problem in-core. In this paper, highly efficient parallel matrix filling schemes are presented for parallel in-core and parallel out-of-core Method of Moments (MoM) integral equation solvers with RWG basis functions. The parallel in-core solver uses RAM, which is often expensive and limited in size. The parallel out-of-core solver is introduced to extend the capability of MoM to solve larger problems which can be as large as the amount of storage on the hard disk. Numerical results on several typical computer platforms show that the parallel matrix filling schemes and matrix equation solvers introduced here are highly efficient and achieve theoretical predictions. The implementation of these advancements with the widely used RWG basis function creates a powerful tool for efficient computational electromagnetic solution of complex real world problems.