•Mathematical formulations for coupled processes in landfills are given.•Each process in landfills is interpreted in detail with different sub-models.•The existing models for simulating coupled MSW behaviors are summarized.•Challenges and perspectives for modeling coupled behaviors of MSW are discussed. Landfilling is the primary method used for municipal solid waste (MSW) disposal. To design, optimize, and manage landfills with a life span of several decades, a deeper understanding of long-term MSW behaviors is necessary and worthwhile. These behaviors should be modeled using approaches that account for coupled processes so as to capture the evolutionary mechanisms that are mainly dominated by biochemical, mechanical, hydraulic, and thermal processes, as well as the complex interactions among them. Many mathematical models have been developed over the past three decades to address this issue. However, most of them only emphasize some of these processes, with only few models accounting for all the processes. In this review, we present a comprehensive overview of the mathematical and numerical formulations of this coupled problem. Each process occurring in landfills is interpreted in detail using different sub-models and the corresponding parameter values. Then, the existing coupled models for MSW are reviewed, and the challenges and perspectives related to the modeling of the long-term behaviors of MSW are highlighted. We conclude that more reliable constitutive formulations based not only on well-designed laboratory tests but also on field tests are necessary to improve the modeling of MSW behaviors in future.