We unify and consolidate various results about non-signall-ing games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of known facts in a variety of special cases. Among these families are {it reflexive games,} which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce {it imitation games,} in which the players display linked behaviour, and which contains as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and {it unique} games. We associate a C*-algebra $C^*(mathcal{G})$ to any imitation game $mathcal{G}$, and show that the existence of perfect quantum commuting (resp. quantum, local) strategies of $mathcal{G}$ can be characterised in terms of properties of this C*-algebra, extending known results about synchronous games. We single out a subclass of imitation games, which we call {it mirror games,} and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra. We describe the main classes of non-signalling correlations in terms of states on operator system tensor products.
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