"Many logicians are familiar with the game theoretic approach to semantics, due to Jaakko Hintikka. This paper by me contains class notes of a logic course at Boston University in fall 1972. It has similar game theoretic ideas, developed quite independently, but influenced by the work of A. Ehrenfeucht. It applies to a larger class of logics, including classical logic, intuitionistic logic and the *-semantics of Ehrenfeucht. The treatment is via D-structures which are finite approximations of infinite structures. For various reasons I did not publish this paper then, but some abstracts, both by myself as well as joint abstracts with Dick De Jongh, Nick Goodman and John Mayberry WERE published in 1972 in the notices of the AMS as well as the Journal of Symbolic Logic. The paper itself finally appeared in the volume to celebrate the 50th birthday of Johan van Benthem."