  This paper is a survey dedicated to the following question: given a group
acting on some CAT(0) cube complex, how to exploit this action to determine
whether or not the group is Gromov / relatively / acylindrically hyperbolic? As
much as possible, the different criteria we mention are illustrated by
applications. We also propose a model for universal acylindrical actions of
cubulable groups and give a few applications to Morse, stable and
hyperbolically embedded subgroups.
