In the paper it is demonstrated, how a dynamic programming approach may be useful for the analysis of Markov games. Markov games with finitely many stages are dealt with extensively. The existence of optimal Markov strategies is proven for finite stage Markov games using a shortcut of a proof by Derman for the analogous result for Markov decision processes. For Markov games with a countably infinite number of stages some results are summarized. Here again the results and the methods of prove have much in common with results and proofs for Markov decision processes. Actually the theory of Markov games is a generalisation. The paper contains short introductions into the theories of matrix games and tree games.