Coevolutionary games involve the simultaneous evolution of strategies and other properties of players, such as their interaction network, reproduction capability, reputation, mobility, or age. This review discusses recent advances in evolutionary games incorporating coevolutionary rules, highlighting their potential to enhance cooperation in social dilemmas. Evolutionary game theory provides a framework for understanding how cooperation emerges in situations where selfish individuals compete. While spatial structure and heterogeneity can promote cooperation, coevolutionary rules can further extend these effects and help explain the emergence of cooperative behavior. Coevolutionary rules allow for the evolution of both strategies and other properties, influencing the interaction network, player reproduction, reputation, mobility, or age. The review covers recent studies on evolutionary games with coevolutionary rules, discusses potential pitfalls and misconceptions, and outlines future research directions. It emphasizes the dynamical effects of coevolutionary rules on the evolution of cooperation, which remain an open area of research. The review also discusses various coevolutionary rules affecting interactions, population growth, teaching activity, mobility, and aging, as well as related aspects of individual and global characteristics that may affect strategy dominance in evolutionary games. The study highlights the importance of coevolutionary rules in promoting cooperation and the need for further research to understand their impact on the evolution of cooperation in social dilemmas.Coevolutionary games involve the simultaneous evolution of strategies and other properties of players, such as their interaction network, reproduction capability, reputation, mobility, or age. This review discusses recent advances in evolutionary games incorporating coevolutionary rules, highlighting their potential to enhance cooperation in social dilemmas. Evolutionary game theory provides a framework for understanding how cooperation emerges in situations where selfish individuals compete. While spatial structure and heterogeneity can promote cooperation, coevolutionary rules can further extend these effects and help explain the emergence of cooperative behavior. Coevolutionary rules allow for the evolution of both strategies and other properties, influencing the interaction network, player reproduction, reputation, mobility, or age. The review covers recent studies on evolutionary games with coevolutionary rules, discusses potential pitfalls and misconceptions, and outlines future research directions. It emphasizes the dynamical effects of coevolutionary rules on the evolution of cooperation, which remain an open area of research. The review also discusses various coevolutionary rules affecting interactions, population growth, teaching activity, mobility, and aging, as well as related aspects of individual and global characteristics that may affect strategy dominance in evolutionary games. The study highlights the importance of coevolutionary rules in promoting cooperation and the need for further research to understand their impact on the evolution of cooperation in social dilemmas.