Advances in software and applications continue to demand advances in memory. The ideal memory would be non-volatile and have maximal capacity, speed, retention time, endurance, and radiation hardness while also having minimal physical size, energy usage, and cost. The programmable metallization cell (PMC) is an emerging memory technology that is likely to surpass flash memory in all the listed ideal memory characteristics. A comprehensive physics-based model is needed to fully understand PMC operation and aid in design optimization. With the intent of advancing the PMC modeling effort, this thesis presents two simulation models for the PMC. The first model is a finite element model based on Silvaco Atlas finite element analysis software. Limitations of the software are identified that make this model inconsistent with the operating mechanism of the PMC. The second model is a physics-based numerical model developed for the PMC. This model is successful in matching data measured from a chalcogenide glass PMC designed and manufactured at ASU. Matched operating characteristics observable in the current and resistance vs. voltage data include the OFF/ON resistances and write/erase and electrodeposition voltage thresholds. Multilevel programming is also explained and demonstrated with the numerical model. The numerical model has already proven useful by revealing some information presented about the operation and characteristics of the PMC.