Abstract Iterated games are well-known in the game theory literature. We study iterated Boolean games. These are games in which players repeatedly choose truth values for Boolean variables they have control over. Our model of iterated Boolean games assumes that players have goals given by formulae of Linear Temporal Logic (LTL), a formalism for expressing properties of state sequences. In order to model the strategies that players use in such games, we use a finite state machine model. After introducing and formally defining iterated Boolean games, we investigate the computational complexity of their associated game-theoretic decision problems as well as semantic conditions characterising classes of LTL properties that are preserved by pure strategy Nash equilibria whenever they exist.