 The algebraizable logics are purported to be the logics with the strongest possible link with their algebraic counterpart. This requirement demands that the algebraic counterpart of the logic should be an algebraic semantics but requires a more robust connection between the logic and the algebraic counterpart than that. This more robust connection is present in the best behaved particular logics known. The mathematically precise concept of algebraizable logic characterizes this type of link. Blok and Pigozzi introduced that fundamental concept in Blok &amp; Pigozzi 1989 and its introduction can be considered the starting point of the unification and growth of the field of abstract algebraic logic in the 1980s. Blok and Pigozzi defined the notion of algebraizable logic only for finitary logics. Later, Czelakowski and Herrmann generalized it to arbitrary logics and also weakened some conditions in the definition. We present here the generalized concept.