  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.
