  This paper develops a theory of conformal density at infinity for groups with
contracting elements. We start by introducing a class of convergence boundary
encompassing many known hyperbolic-like boundaries, on which a detailed study
of conical points and Myrberg points is carried out. The basic theory of
conformal density is then established, including the Sullivan shadow lemma and
Hopf-Tsuji-Sullivan theorem. This gives a unification of the theory of
conformal density on the Gromov and Floyd boundary for (relatively) hyperbolic
groups, the visual boundary for rank-1 CAT(0) groups, and Thurston boundary for
mapping class groups. Besides that, the conformal density on the horofunction
boundary provides a new important example of our general theory. At last, we
present applications towards the co-growth problem of divergent groups, measure
theoretical results for CAT(0) groups and mapping class groups.
