For any group G, I will define the set of hyperbolic structures on G, denoted H(G), which consists of equivalence classes of (possibly infinite) generating sets of G such that the corresponding Cayley graph is hyperbolic. Alternatively, one can define hyperbolic structures in terms of cobounded G-actions on hyperbolic spaces. Of special interest is the subset AH(G) of H(G) , which consists of acylindrically hyperbolic structures on G, i.e. hyperbolic structures corresponding to acylindrical actions. I will discuss basic properties of these posets such as cardinality, existence of extremal elements, results about hyperbolic structures induced from hyperbolically embedded subgroups of G and accessibility. Lastly I will talk about some recent work regarding quasi-parabolic structures.