This work is the result of a wide range of computer simulation research into the aggregation behavior of dispersed colloidal and aerosol particles in a number of different environments from the continuum to the free-molecular. The goal of this research has been to provide a bridge between experimental and theoretical researchers in this field by simulating the aggregation process within a known model. To this end, a variety of interparticle interactions has been studied in the course of this research, focusing on the effect of these interactions on the aggregation mechanism and resulting aggregate structures. Both Monte Carlo and Brownian Dynamics codes have been used to achieve this goal. The morphologies of clusters that result from aggregation events in these systems have been thoroughly analyzed with a range of diverse techniques, and excellent agreement has been found with other researchers in this field. Morphologies of these clusters include fractal, gel, and crystalline forms, sometimes within the same structure at different length scales. This research has contributed to the fundamental understanding of aggregation rates and size distributions in many physical system, having allowed for the development of improved models of the aggregation and gelation process. Systems studied include DLCA and BLCA in two and three dimension, free-molecular diffusional (Epstein) system, selective aggregation in binary colloids, ssDNA mediated aggregation in colloidal systems, and several others.