The theory of semihypergroups is a natural extension to the theory of locally compact semigroups. In this article, we present different notions of amenability, namely amenability of function-spaces and topological amenability, in the broader setting of (locally compact) semihypergroups and survey some recent developments in this area of research regarding certain ergodic, stationary, hereditary, Banach algebraic and fixed-point characterizations of such notions on general (semitopological) semihypergroups.