rivin conjectured that the formal conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. in this talk, i will present a proof of rivin's conjecture and supporting evidence for the analogous statement for acylindrically hyperbolic groups. the class of acylindrically hyperbolic groups is a wide class of groups that contains (among many other examples) the outer automorphism groups of free groups and the mapping class groups of hyperbolic surfaces. this is a joint work with laura ciobanu. for the pre-talk: hyperbolic groups will be defined and it will be explained why the generating function of the sequence counting the number of elements of length n is rational.