Heterogeneous systems are present in a variety of scientific and engineering fields including biology, structural engineering, aerospace systems, material science, and geology, just to name a few. Many of these materials arrange randomly with locally complex features and only lend themselves to being characterized statistically. Of particular interest to this work is determining effective (macroscopic) transport and mechanical properties of particulate composites from higher order statistical data, which is a fundamental problem studied by many including Einstein and Maxwell. The geometrical arrangement of heterogeneous materials plays a critical role in a material's behavior, and this work illustrates that the detailed characterization of real materials is essential to accurately predict the macroscopic (overall/effective) behavior. The past half century has seen the formation of rigorous bounds and ap- proximate theories relying on a higher order description of the complex morphologies. However, assumptions are often made about the microstructure including alignment of inclusions, dispersity of particle sizes, etc., and inclusions in real materials are often irregularly shaped and can lead to anisotropic and complex configurations. Therefore, the impact of these arrangements on material behavior is not well understood. In order to limit assumptions made about the microstructural features, real materials are characterized in this work using micro-computer tomography, and a framework is developed for determining microstructure-statistics-property relations for a wide spectrum of microstructures. In this work, anisotropic second-order and isotropic third-order statistical micromechanics estimates are obtained for complex highly packed microstructures regardless of the complexity of the probability spectrum. This is accomplished through development of adaptive integration techniques and a Monte Carlo statistical sampling algorithm designed for a high performance computing environment. Verification and validation studies are conducted, and good agreement is shown with direct numerical modeling. Moreover, these methods are able to predict effective behavior for particle concentrations and infinite material contrast ratio beyond the limits of direct numerical approaches. For the first time, anisotropic second order estimates are computed for highly packed systems of non- spherical inclusions. In addition, third-order estimates predict a significant shape effect for the effective thermal conductivity and a moderate shape effect for elastic constants when considering composites with Platonic solid inclusions.