In this talk we introudce McNaughton games played on finite graphs. These games present natural objects of study from computational complexity point of view, and can be used as models of reactive systems, communication and update networks and processes of infinite duruation. Using these games one can address issues on realizability of specifications, correct program synthesis and optimization of control for (reactive) systems. In this talk we discuss some examples of McNaughton games and (if time permits) present the relationship between the structure of the underlying graphs, complexity of winning strategies and algorithms for deciding the winners of the games considered.