Evolutionary game dynamics applies population dynamical methods to game theory, focusing on how strategy frequencies in populations change over time based on their success. Introduced by evolutionary biologists, it builds on classical game theory concepts. This survey covers various deterministic dynamical systems inspired by evolutionary game theory, including ordinary differential equations (like the replicator equation), differential inclusions (best response dynamics), difference equations (fictitious play), and reaction-diffusion systems. A key theme is the connection between dynamical approaches and Nash equilibria, though static equilibrium-based views may not always capture long-term behavior. The replicator equation models how successful strategies spread in a population. It describes the dynamics of strategy frequencies based on payoffs. Nash equilibria are closely related to rest points of the replicator equation. A rest point is a strategy that is a best reply to itself, and strict Nash equilibria are those where no other strategy can invade. The replicator equation's rest points correspond to Nash equilibria, with strict Nash equilibria being asymptotically stable. The classification of phase portraits for the replicator equation reveals different dynamics depending on the number of strategies. For n=2, the equation reduces to a one-dimensional system with possible stable or unstable equilibria. For n=3, Zeeman and Bomze classified phase portraits, showing the existence of unique rest points and the possibility of bistability. For n=4, the dynamics are more complex, with periodic and chaotic attractors. The replicator equation is permanent if there exists a compact set within the interior of the simplex that all orbits eventually enter. This implies that strategies persist despite random shocks. Theorems show that permanence is related to the existence of a unique rest point and the positivity of certain determinants. Mixed strategy dynamics and evolutionary stable strategies (ESS) are central to evolutionary game theory. An ESS is a strategy that cannot be invaded by any other strategy, even if it starts with a small frequency. The conditions for an ESS involve the strategy being a Nash equilibrium and having a strict stability condition. Theorems confirm that ESS are asymptotically stable rest points and that they correspond to strong stability in the population dynamics. The relationship between evolutionary and dynamic stability is particularly clear in partnership games, where both players' interests align. In such games, an ESS is asymptotically stable. For other games, ES sets generalize the concept of ESS, allowing for sets of strategies that are neutrally stable. These concepts highlight the importance of dynamic stability in understanding long-term evolutionary outcomes.Evolutionary game dynamics applies population dynamical methods to game theory, focusing on how strategy frequencies in populations change over time based on their success. Introduced by evolutionary biologists, it builds on classical game theory concepts. This survey covers various deterministic dynamical systems inspired by evolutionary game theory, including ordinary differential equations (like the replicator equation), differential inclusions (best response dynamics), difference equations (fictitious play), and reaction-diffusion systems. A key theme is the connection between dynamical approaches and Nash equilibria, though static equilibrium-based views may not always capture long-term behavior. The replicator equation models how successful strategies spread in a population. It describes the dynamics of strategy frequencies based on payoffs. Nash equilibria are closely related to rest points of the replicator equation. A rest point is a strategy that is a best reply to itself, and strict Nash equilibria are those where no other strategy can invade. The replicator equation's rest points correspond to Nash equilibria, with strict Nash equilibria being asymptotically stable. The classification of phase portraits for the replicator equation reveals different dynamics depending on the number of strategies. For n=2, the equation reduces to a one-dimensional system with possible stable or unstable equilibria. For n=3, Zeeman and Bomze classified phase portraits, showing the existence of unique rest points and the possibility of bistability. For n=4, the dynamics are more complex, with periodic and chaotic attractors. The replicator equation is permanent if there exists a compact set within the interior of the simplex that all orbits eventually enter. This implies that strategies persist despite random shocks. Theorems show that permanence is related to the existence of a unique rest point and the positivity of certain determinants. Mixed strategy dynamics and evolutionary stable strategies (ESS) are central to evolutionary game theory. An ESS is a strategy that cannot be invaded by any other strategy, even if it starts with a small frequency. The conditions for an ESS involve the strategy being a Nash equilibrium and having a strict stability condition. Theorems confirm that ESS are asymptotically stable rest points and that they correspond to strong stability in the population dynamics. The relationship between evolutionary and dynamic stability is particularly clear in partnership games, where both players' interests align. In such games, an ESS is asymptotically stable. For other games, ES sets generalize the concept of ESS, allowing for sets of strategies that are neutrally stable. These concepts highlight the importance of dynamic stability in understanding long-term evolutionary outcomes.