Abstract: We consider the basic concepts, such as players, strategies, payoff functions, dominance, equilibrium situations for noncooperative games. For antagonistic matrix games the concept of saddle point and the minimax theorem are used to show the existence of equilibrium situations in the set of mixed strategies. The famous Nash theorem about the existence of equilibrium situations in the set of mixed strategies will be presented. Finally, we discuss the problem of fair payoffs for cooperative games with finitely many players, which amount to the existence of the Shapley vector.