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.