The introduction of the approximate subdifferential of a convex function has been proved to be useful in convex optimization, from the theoretical viewpoint as well as for the purposes of devising algorithms. Within the context of necessary and sufficient conditions for almost optimality, it was known from the beginning that to claim that a point x ° is an ~-minimum of f is equivalent to declaring that 0 belongs to the ~-subdifferential of f at x o. From the point of view of minimization procedures, it iswidely recognized that ~-subgradient methods can be more usable than methods using exact subgradients. That is mainly due to the fact that it is often easier to have access to an ~-subgradient than to a subgradient. For what concerns the study and the use of ~-subdifferentials, the past sixteen years can roughly be divided into three periods of time :