In this paper we present a review of the most important notions and characterizations of differentiability and necessary optimality conditions for a fuzzy multiobjective problem. As basis of this review, we first study the fundamental aspects of the notions of differentiability for interval valued functions, since the fuzzy environment and the interval environment are closely related. Those aspects are related to the different definitions of difference for intervals and their drawbacks, the different definitions and characterizations of the differentiability for interval-valued functions and their drawbacks and how they have been solved in the literature. Based on the most important and meaning results on interval valued functions you can find in the literature, a review on notions of differentiability in fuzzy context is given, both in the case of functions of one variable, and several variables. And finally we present the review results of the necessary optimality conditions for fuzzy multiobjective problems and the main conclusions.