This review paper provides the operations management (OM) community with an exhaustive analysis of the mathematical models developed for the problem of low-carbon supply chain management (LCSCM). Our paper belongs to the green supply chain management (GSCM) reviews but is distinguished by its specific interest in analysing research works on supply chain (SC) management regarding the reduction of carbon emissions and its related constraints. To facilitate our benchmarking of the 83 selected papers, we adopt a literature classification based on the logistic decisions studied within the developed models. We distinguish three categories of logistic decisions: operational management, technology investment and SC design coordination. Companies are currently facing great external pressures from governments and their conscientious customers to reduce their overall emissions. We analyse how these environmental constraints, which we believe are key drivers for low-carbon emissions management, have been incorporated into mathematical models. Analysing these external pressures in terms of concern about carbon emissions constitutes our main contribution through this literature review. In addition, companies are facing a challenge to reduce their carbon emissions, which are mainly generated from production, transport and storage activities. Consequently, the modelling of carbon emissions remains a crucial task when addressing the LCSCM problem. We suggest analysing the techniques used thus far to approximate those carbon emissions. Furthermore, to illustrate our literature classification and the features of the LCSCM problem, we provide the framework on which we based our analysis of the selected literature. We discuss the modelling aspects of this problem to highlight the limits of the existing literature and consequently suggest recommendations for future research. We believe that this issue will continue to be one of the top concerns of the OM community within the GSCM field as it continues to gain importance among business leaders, and political and social actors.