Microstructural morphology and grain boundary properties often control the service properties of engineered materials. This report uses the Potts-model to simulate the development of microstructures in realistic materials. Three areas of microstructural morphology simulations were studied. They include the development of massively parallel algorithms for Potts-model grain grow simulations, modeling of mass transport via diffusion in these simulated microstructures, and the development of a gradient-dependent Hamiltonian to simulate columnar grain growth. Potts grain growth models for massively parallel supercomputers were developed for the conventional Potts-model in both two and three dimensions. Simulations using these parallel codes showed self similar grain growth and no finite size effects for previously unapproachable large scale problems. In addition, new enhancements to the conventional Metropolis algorithm used in the Potts-model were developed to accelerate the calculations. These techniques enable both the sequential and parallel algorithms to run faster and use essentially an infinite number of grain orientation values to avoid non-physical grain coalescence events. Mass transport phenomena in polycrystalline materials were studied in two dimensions using numerical diffusion techniques on microstructures generated using the Potts-model. The results of the mass transport modeling showed excellent quantitative agreement with one dimensional diffusion problems, however the results also suggest that transient multi-dimension diffusion effects cannot be parameterized as the product of the grain boundary diffusion coefficient and the grain boundary width. Instead, both properties are required. Gradient-dependent grain growth mechanisms were included in the Potts-model by adding an extra term to the Hamiltonian. Under normal grain growth, the primary driving term is the curvature of the grain boundary, which is included in the standard Potts-model Hamiltonian.