Minimax problems play an important role in different fields of nonlinear analysis (game theory, duality theory, fixed point theory). After the introduction of the problem and some of its basic properties the paper investigates penalty methods to solve minimax problems. Similar to ordinary optimization problems the author distinguished between interior and exterior methods, analyzed in sections 2 and 3, respectively. The exterior methods have the advantage that they are easy to start, but often yield unfeasible solutions. To overcome this drawback the author proposes in the third section an extension to these methods. The paper ends with considerations for min-sup problems, i.e. for the case of the determination of half saddle points.