This review summarizes recent advances in the study of evolutionary dynamics of group interactions on structured populations, including lattices, complex networks, and coevolutionary models. It compares these results with those obtained on well-mixed populations. The review highlights the importance of statistical physics, network science, and evolutionary game theory in understanding the dynamics of group interactions, which are crucial for biological, economic, and social sciences. The focus is on the public goods game, a representative of group interactions, and its evolution on structured populations. The review discusses the effects of group size, heterogeneity in pay-offs and contributions, and strategic complexity on the evolution of cooperation. It also explores the role of social heterogeneity, the impact of network structure, and the effects of punishment and reward on the evolution of cooperation. The review emphasizes the importance of structured populations in promoting cooperation and the need for further research in this area. The study shows that structured populations can lead to the emergence of cooperative clusters and the evolution of public cooperation, which is not observed in well-mixed populations. The review also discusses the challenges and pitfalls in simulating evolutionary games on structured populations and the importance of using appropriate methods to ensure accurate results. Overall, the review provides a comprehensive overview of the current state of research on evolutionary dynamics of group interactions on structured populations and highlights the potential of these models in understanding complex systems.This review summarizes recent advances in the study of evolutionary dynamics of group interactions on structured populations, including lattices, complex networks, and coevolutionary models. It compares these results with those obtained on well-mixed populations. The review highlights the importance of statistical physics, network science, and evolutionary game theory in understanding the dynamics of group interactions, which are crucial for biological, economic, and social sciences. The focus is on the public goods game, a representative of group interactions, and its evolution on structured populations. The review discusses the effects of group size, heterogeneity in pay-offs and contributions, and strategic complexity on the evolution of cooperation. It also explores the role of social heterogeneity, the impact of network structure, and the effects of punishment and reward on the evolution of cooperation. The review emphasizes the importance of structured populations in promoting cooperation and the need for further research in this area. The study shows that structured populations can lead to the emergence of cooperative clusters and the evolution of public cooperation, which is not observed in well-mixed populations. The review also discusses the challenges and pitfalls in simulating evolutionary games on structured populations and the importance of using appropriate methods to ensure accurate results. Overall, the review provides a comprehensive overview of the current state of research on evolutionary dynamics of group interactions on structured populations and highlights the potential of these models in understanding complex systems.