In the modern literature on game theory there are several versions of what is known as Zermelo's theorem. It is shown that most of these modern statements of Zermelo's theorem bear only a partial relationship to what Zermelo really did. We also give a short survey and discussion of the closely related but almost unknown work by Konig and Kalmar. Their papers extend and considerably generalize Zermelo's approach. A translation of Zermelo's paper is included in the appendix.