For every group G, we introduce the set of acylindrically hyperbolic structures on G, denoted AH(G). One can think of elements of AH(G) as cobounded acylindrical G-actions on hyperbolic spaces considered up to a natural equivalence.Elements of AH(G) can be ordered in a natural way according to the amount of information they provide about the group G. We will discuss some basic questions about the poset structure of AH(G) as well as more advanced results about the existence of maximal acylindrically hyperbolic structures and rigidity phenomena.