For finitely generated groups, growth of the elements of the group, and the series (or generating functions) associated to the growth function, have been widely studied. Recently researches have begun to study the growth of conjugacy classes in these groups. Disconcertingly, the conjugacy growth series had been found by Rivin not to be rational for free groups with respect to a free basis. In this talk I will introduce the notion of geodesic conjugacy growth functions and series, and discuss the effects of various group constructions on rationality of both the geodesic conjugacy and (spherical) conjugacy languages whose growth is measured by these functions. In particular, we show that rationality of the geodesic conjugacy growth series, as well as on regularity of the geodesic and spherical conjugacy growth series is preserved by both direct and free products. This is joint work with Laura Ciobanu.