The notion of a convergence group first appeared in the work of Furstenberg, under the name "Dynkin group". The well-known terminology "convergence group" is due to Gehring-Martin. Typical examples of convergence groups include hyperbolic and relatively hyperbolic groups but in general a convergence group is not necessarily relatively hyperbolic. Yet we prove that non-elementary convergence groups satisfy a generalization of relative hyperbolicity called acylindrical hyperbolicity, which allows the theory of acylindrically hyperbolic groups to be applied to the study of convergence groups. Moreover, we define a generalized convergence group which gives a dynamical characterization of acylindrical hyperbolicity. Speaker(s): Bin Sun (Oxford)