Hyperbolicities in CAT(0) cube complexes

This paper is a survey dedicated to the following question: given a group acting on a 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.