Particle swarm optimization (PSO) is a well-known stochastic population-based search algorithm, originally developed by Kennedy and Eberhart in 1995. Given PSO's success at solving numerous real world problems, a large number of PSO variants have been proposed. However, unlike the original PSO, most variants currently have little to no existing theoretical results. This lack of a theoretical underpinning makes it difficult, if not impossible, for practitioners to make informed decisions about the algorithmic setup. This thesis focuses on the criteria needed for particle stability, or as it is often refereed to as, particle convergence. While new PSO variants are proposed at a rapid rate, the theoretical analysis often takes substantially longer to emerge, if at all. In some situation the theoretical analysis is not performed as the mathematical models needed to actually represent the PSO variants become too complex or contain intractable subproblems. It is for this reason that a rapid means of determining approximate stability criteria that does not require complex mathematical modeling is needed. This thesis presents an empirical approach for determining the stability criteria for PSO variants. This approach is designed to provide a real world depiction of particle stability by imposing absolutely no simplifying assumption on the underlying PSO variant being investigated. This approach is utilized to identify a number of previously unknown stability criteria. This thesis also contains novel theoretical derivations of the stability criteria for both the fully informed PSO and the unified PSO. The theoretical models are then empirically validated utilizing the aforementioned empirical approach in an assumption free context. The thesis closes with a substantial theoretical extension of current PSO stability research. It is common practice within the existing theoretical PSO research to assume that, in the simplest case, the personal and neighborhood best positions are stagnant. However, in this thesis, stability criteria are derived under a mathematical model where by the personal best and neighborhood best positions are treated as convergent sequences of random variables. It is also proved that, in order to derive stability criteria, no weaker assumption on the behavior of the personal and neighborhood best positions can be made. The theoretical extension presented caters for a large range of PSO variants.