The fuzzy theory is a generalization of the standard set theory that is based on the membership function, which expresses, in the fuzzy sense, the membership degree of an element to a set. After a review of the main operations on fuzzy sets, some applications are proposed in various contexts such as in engineering sciences or computational sciences, in automatic systems control or quality evaluation. Special attention is devoted to the applications of fuzzy theory in cognitive sciences by highlighting various critical issues reported in the literature and some responses to these. The intuitionistic and hesitant settings are then introduced and it is shown how these operate the union and intersection of fuzzy sets. In the intuitionistic fuzzy theory; along with a membership function, a non-membership function is defined and uncertainty is modeled. Besides, the hesitant fuzzy theory allows to express the uncertainty of one or more decision makers.