Anisotropy in the spatial arrangement of species and also geometric anisotropy in the components of liquid and soft matter systems gives rise to complex phase and non-trivial dynamic behaviors. Many systems are encompassed by this description including polymers, guest molecules in porous gels or nano-structured materials and also orientable fluids. In order to understand these systems, computational studies provide valuable insight. The study of such systems computationally is difficult if not prohibitive thus it is necessary to reduce these systems to simple models that capture the essential physical processes that govern the dynamics and phase behavior. Two simple models fit into this category: systems where the surrounding solvent is held stationary, a Lorentz model and also systems where a liquid crystal is formed from hard spherocylinders and is driven by an electric field. The Lorentz models studied provide a description of the dynamical regimes accessible when the probe and/or scatterers are given geometric anisotropy. The resulting dynamics are studied when order is present in the stationary solvent, i.e. structured versus isotropic solvent structure. Effective channels depending upon the competition of length scales as well as the structure of the stationary solvent leads to transitions in the dynamics of the traversing probe. Geometric anisotropy in nematogens in liquid crystals leads to the formation of mesophases indicative of a liquid crystal. When coupled to a rotating electric field, the dynamics and phase of the nematogens can be controlled.