The opioid overdose crisis is driven by an intersecting set of social, structural, and economic forces. Simulation models offer a tool to help us understand and address this complex, dynamic, nonlinear, social phenomenon. We conducted a systematic review of the literature on simulation models of opioid use and overdose up to September 2019. We extracted modeling types, target populations, interventions, and findings. Further, we created a database of model parameters used for model calibration, and evaluated study transparency and reproducibility. Of the 1,381 articles screened, we identified 72 eligible articles. The most frequent types of models were Markov (26%), compartmental (25%), system dynamics (19%), and Agent-Based models (18%). Almost half (46%) evaluated intervention cost-effectiveness, while 29% of studies focused on treatment and harm reduction services for people with opioid use disorder (OUD). More than half (57%) calibrated their models to empirical data, and 31% discussed validation approaches used in their modeling process. From the 51 studies that provided data on model parameters, we mapped out the data sources for parameters on opioid use, OUD, OUD treatment, cessation/relapse, emergency medical services, and mortality. This database offers a tool that future modelers can use to identify model inputs, and to evaluate comparability of their models to prior work. Future applications of simulation models to this field should actively tackle key methodological challenges, including the potential for bias in the choice of parameter inputs, investment in model calibration and validation, and transparency in the assumptions and mechanics of simulation models to facilitate reproducibility.