Background: Many open source software (OSS) quality assessment models are proposed and available in the literature. However, there is little or no adoption of these models in practice. In order to guide the formulation of newer models so they can be acceptable by practitioners, there is need for clear discrimination of the existing models based on their specific properties. Based on this, the aim of this study is to perform a systematic literature review to investigate the properties of the existing OSS quality assessment models by classifying them with respect to their quality characteristics, the methodology they use for assessment, and their domain of application so as to guide the formulation and development of newer models. Searches in IEEE Xplore, ACM, Science Direct, Springer and Google Search is performed so as to retrieve all relevant primary studies in this regard. Journal and conference papers between the year 2003 and 2015 were considered since the first known OSS quality model emerged in 2003. Results: A total of 19 OSS quality assessment model papers were selected. To select these models we have developed assessment criteria to evaluate the quality of the existing studies. Quality assessment models are classified into five categories based on the quality characteristics they possess namely: single-attribute, rounded category, community-only attribute, non-community attribute as well as the non-quality in use models. Our study reflects that software selection based on hierarchical structures is found to be the most popular selection method in the existing OSS quality assessment models. Furthermore, we found that majority (47%) of the existing models do not specify any domain of application. Conclusions: In conclusion, our study will be a valuable contribution to the community and helps the quality assessment model developers in formulating newer models and also to the practitioners (software evaluators) in selecting suitable OSS in the midst of alternatives.