In 1970s, Grigory Margulis established the asymptotic formulae for the number of closed geodesics on compact Riemannian manifolds with negative curvature. In this talk, we will discuss the analog of the question in the setting of groups: count the number of conjugacy classes. One result is that the number of conjugacy classes in relatively hyperbolic groups has a similar growth pattern as the manifold case. The same result is also obtained for CAT(0) groups with a rank-1 element. In fact, these results can be stated in a general class of groups with contracting elements. As a consequence of this formulae, the Rivin conjecture holds for these groups that the generating function for conjugacy classes is transcendental.海报