Uninorms are a generalization of t-norm and t-conorm operators commonly used to interpret ands'' and ors’’ in fuzzy logics i.e. commutative, associative, increasing binary functions on the real unit interval [0,1], with an identity element e that can appear anywhere in the interval (e=1 giving a t-norm, e=0 giving a t-conorm). The idea of this talk is to generalize the well-known construction of fuzzy logics based on t-norms to fuzzy logics based on uninorms. To this end we give an axiomatization for a new basic fuzzy logic U which can be viewed both as Monoidal t-norm based fuzzy logic MTL without weakening, and as a kind of ``linear linear logic’’. We then investigate axiomatic extensions of U which correspond to logics based on particular classes of uninorms, including a logic of idempotent uninorms which turns out to be the relevant logic RM, and a logic of continuous uninorms related to the combining function of the expert system MYCIN.