The authors present an interesting survey of predicate fuzzy logics based on triangular norms (t-norms), collecting previous results by themselves and by other authors in the more general context of core fuzzy logics (i.e., logics expanding MTL and satisfying the local deduction theorem) and Delta-core fuzzy logics (i.e., logics expanding MTL with Delta and satisfying the Delta deduction theorem). Besides the main definitions and results in the general case, some variants are also presented, for example the logic with crisp equality and the logic of witnessed models. Complexity issues and some model theory are also dealt with. The paper ends with a section on examples of theories over predicate fuzzy logic (such as fuzzy arithmetic and fuzzy set theory) and with a very complete list of references.