The main classification of sentential logics in Abstract Algebraic Logic (AAL) is the so-called Leibniz hierarchy. It has its roots in the seminal work of Blok and Pigozzi on algebraizable logics, and it classifies a given logic according to algebraic properties of the Leibniz operator over the logical filters. In this talk we shall focus on the less-known Suszko operator, introduced by Czelakowski, which seems to play a more significant rôle when dealing with non-protoalgebraic logics. We present some recent developments in AAL arising from this shifting of paradigm-operator. Namely, we characterize the main classes of logics within the Leibniz hierarchy through algebraic properties of the Suszko operator, and present new isomorphism theorems for protoalgebraic and equivalential logics, in the same spirit of the known ones for weakly-algebraizable and algebraizable logics. This is a joint work with Ramon Jansana and Josep Maria Font.