This thesis covers the main results given by the evolutionary game theory. In this approach, the players are individuals from the same population and the competition between pairs of randomly drawn individuals is studied. The probabilities in mixed strategies represent the population shares of a given type of individual. The solution of this kind of game is the evolutionary stable strategy, that is, the strategy that is resilient to the appearance of mutants in the population. Evolutionary stability requires that the individuals that follow the incumbent strategy get the higher payoff than the mutants (the individuals who started following some other strategy) as long as the share of mutants in the population is sufficiently small. Although this approach doesn’t require rationality, it turns out that every evolutionary stable strategy is also in Nash equilibrium with itself. In the first part some more properties of the evolutionary stable strategies are described and the equivalence between evolutionary stable and socially optimal strategies in doubly symmetric games is shown. The second half covers how the population state changes over time. With the assumption that the population share of individuals that achieve better than average payoff increases, the replicator dynamics for each type of individual is derived. Furthermore, it is shown that strictly dominated strategies vanish from the population in the long run, while that is not necessarily so for the weakly dominated strategies. After that, it is shown that Nash equilibria corresponds to the stationary states of the replicatory dynamics and that it’s necessary for the state to be a Nash equilibrium for it to be stable. In the end, it is shown that evolutionary stability is the necessary condition for asymptotic stability and that with doubly symmetric games the equivalence holds.