Convex functions and their analogues have been powerful tools in almost all mathematical fields, including optimization, fractional calculus, mathematical analysis, functional analysis, operator theory, and mathematical physics. It is well established in the literature that a convex function with is necessarily superadditive, while a concave function is subadditive. The converses of these two assertions are not valid in general. The main target of this article is to study the subadditivity and superadditivity of convex and superquadratic functions. In particular, we obtain several results extending, refining, and reversing some known inequalities in this direction. Further discussion of superquadratic functions in this line will be given.