This review provides a tutorial-type overview of evolutionary games on graphs for physicists. It begins by introducing classical and evolutionary game theory, covering definitions, Nash equilibrium, potential and zero-sum games, and multi-player games. The review then discusses the dynamics of three prominent game classes: the Prisoner's Dilemma, Rock-Scissors-Paper, and Competing Associations. The main theme is how the graph structure of interactions can modify and enrich the long-term behavioral patterns in evolutionary games. The review also explores the topological complications of non-mean-field social network structures and the impact of agent-based dynamics on game outcomes. Finally, it concludes with a discussion of open questions and future research directions.This review provides a tutorial-type overview of evolutionary games on graphs for physicists. It begins by introducing classical and evolutionary game theory, covering definitions, Nash equilibrium, potential and zero-sum games, and multi-player games. The review then discusses the dynamics of three prominent game classes: the Prisoner's Dilemma, Rock-Scissors-Paper, and Competing Associations. The main theme is how the graph structure of interactions can modify and enrich the long-term behavioral patterns in evolutionary games. The review also explores the topological complications of non-mean-field social network structures and the impact of agent-based dynamics on game outcomes. Finally, it concludes with a discussion of open questions and future research directions.