In this talk we discuss a new class of non-signalling games, which we call imitation games, and which includes as subclasses the class of synchronous (and hence graph-colouring and graph homomorphism) games, the class of binary constraint system games, and the class of unique games, among others. We attach a C*-algebra to each imitation game, and show that the winning non-signalling strategies for the game, belonging to some fixed classes, e.g. that of quantum commuting correlations, are in a one-to-one correspondence with traces on the game C*-algebra. We will recall the connection between winning strategies for general non-signalling games and states on tensor products of operator systems, and will show that, in the case the non-signalling game is reflexive (a class of games to be introduced in the talk), its winning strategies can be described via states on quotient operator systems.