his talk presents various characterizations of the subdifferential of the pointwise supremum of an arbitrary family of convex functions, as well as some featured applications. Starting by the maximum generality framework, we move after to particular contexts in which some continuity and compacity assumptions are either imposed or inforced via processes of compactification of the index set and regularization of the data functions. Some relevant applications of the general results are presented, in particular to derive rules for the subdifferential of the sum, and for convexifying a general (unconstrained) optimization problem. The last part deals with some specific constraint qualifications for the convex optimization problem with an arbitrary set of constraints, and also contains different sets of KKT-type optimality conditions appealing to the subdifferential of the supremum function.

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