Aging has a serious impact on the properties of functional polymers. Therefore, it is necessary to study the aging mechanism to prolong the service and storage life of polymer-based devices and materials. Due to the limitations of traditional experimental methods, more and more studies have adopted molecular simulations to analyze the intrinsic mechanisms of aging. In this paper, recent advances in molecular simulations of the aging of polymers and their composites are reviewed. The characteristics and applications of commonly used simulation methods in the study of the aging mechanisms (traditional molecular dynamics simulation, quantum mechanics, and reactive molecular dynamics simulation) are outlined. The current simulation research progress of physical aging, aging under mechanical stress, thermal aging, hydrothermal aging, thermo-oxidative aging, electric aging, aging under high-energy particle impact, and radiation aging is introduced in detail. Finally, the current research status of the aging simulations of polymers and their composites is summarized, and the future development trend has been prospected.