Evolutionary games on graphs, as discussed in this review, explore how the structure of interactions among agents influences long-term behavioral patterns in non-cooperative games. The paper provides a tutorial overview for physicists, covering classical and evolutionary game theory, the dynamics of three prominent game classes (Prisoner's Dilemma, Rock-Scissors-Paper, and Competing Associations), and the implications of non-mean-field social network structures. It emphasizes how graph structures can modify and enrich the outcomes of evolutionary games, particularly in terms of cooperation and stability. The review begins by introducing rational game theory, focusing on Nash equilibrium, social dilemmas, and potential games. It then transitions to evolutionary game theory, discussing population dynamics, agent-based models, and the role of graph structures in shaping outcomes. The paper highlights the importance of considering the topology of social networks, such as lattices, small worlds, and scale-free graphs, in understanding how interactions influence game outcomes. Key findings include the emergence of cooperation in spatial games, the role of network structure in stabilizing or destabilizing equilibria, and the importance of bounded rationality in explaining real-world behavior. The review also discusses the limitations of classical game theory, particularly in modeling situations with cognitive constraints and non-trivial network structures. It emphasizes the need for dynamic models that account for the evolution of strategies and the role of stochasticity in shaping long-term outcomes. The paper concludes by outlining future research directions, including the application of evolutionary game theory to complex systems, the development of new analytical tools, and the exploration of non-equilibrium statistical physics in understanding evolutionary dynamics. It also highlights the importance of considering both strategic and structural heterogeneity in social networks to better understand the emergence of cooperative behavior and the stability of equilibria in evolutionary games.Evolutionary games on graphs, as discussed in this review, explore how the structure of interactions among agents influences long-term behavioral patterns in non-cooperative games. The paper provides a tutorial overview for physicists, covering classical and evolutionary game theory, the dynamics of three prominent game classes (Prisoner's Dilemma, Rock-Scissors-Paper, and Competing Associations), and the implications of non-mean-field social network structures. It emphasizes how graph structures can modify and enrich the outcomes of evolutionary games, particularly in terms of cooperation and stability. The review begins by introducing rational game theory, focusing on Nash equilibrium, social dilemmas, and potential games. It then transitions to evolutionary game theory, discussing population dynamics, agent-based models, and the role of graph structures in shaping outcomes. The paper highlights the importance of considering the topology of social networks, such as lattices, small worlds, and scale-free graphs, in understanding how interactions influence game outcomes. Key findings include the emergence of cooperation in spatial games, the role of network structure in stabilizing or destabilizing equilibria, and the importance of bounded rationality in explaining real-world behavior. The review also discusses the limitations of classical game theory, particularly in modeling situations with cognitive constraints and non-trivial network structures. It emphasizes the need for dynamic models that account for the evolution of strategies and the role of stochasticity in shaping long-term outcomes. The paper concludes by outlining future research directions, including the application of evolutionary game theory to complex systems, the development of new analytical tools, and the exploration of non-equilibrium statistical physics in understanding evolutionary dynamics. It also highlights the importance of considering both strategic and structural heterogeneity in social networks to better understand the emergence of cooperative behavior and the stability of equilibria in evolutionary games.