A sustainable transport infrastructure is one of the pillars of a sustainable city. However, the literature indicates that urbanization, population growth, changes in population density, and motorization make it difficult for the current road transport system to meet mobility needs for a sustainable city. Traffic crashes and congestion on roads are common as a result of increasing travel times, fuel consumption, and carbon emissions, thereby reducing efficiency and sustainability of mobility systems. Managing these issues involves the interaction of multiple decision-makers, such as vehicles, pedestrians, traffic system operators, and authorities. Accordingly, these are well-suited to being analyzed under the guise of game theory. While classical game theory possesses multiple limitations, it can be argued that evolutionary game theory (EGT) models are more effective for real-world scenarios. This manuscript presents a state-of-the-art review on EGT applied to the road transportation network. The manuscript has divided the application of EGT in advancing the transportation network into multiple categories, i.e., choice-based analysis, traffic management, behavioral interactions, routing operation, and transport safety. This manuscript provides an in-depth analysis and a comparative criticism of the various proposed evolutionary game models. Finally, the manuscript discusses the challenges and provides recommendations for future research on evolutionary game models in transportation networks. These insights aim to facilitate targeted activities based on current research needs.