In this talk I will give a light survey on some recent works on bisychronous correlations and bisynchronous games. Bisynchronous correlations form a special class of synchronous correlations, where there is the additional constraint that the probability of Alice and Bob giving the same outputs whenever they receive distinct inputs is always zero. I will explain how bisynchronous correlations are connected to traces on quantum permutation groups, and how bisynchronous correlations and bisynchronous games fit within the context of MIP*=RE. Time permitting I'll also explain how for certain classes of bisynchronous games (e.g., graph isomorphism games), one can identify perfect strategies using purely combinatorial-algebraic methods by studying representation categories of certain quantum automorphism groups.