ABSTRACT:  The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, allowing us in principle to logically separate the notions of contradictoriness and of inconsistency.  We present the formal definitions of these logics in the context of General Abstract Logics, argue that they in fact represent the majority of all paraconsistent logics existing up to this point, if not the most exceptional ones, and we single out a subclass of them called C-systems, as the LFIs that are built over some given positive basis.  Given precise characterizations of some received logical principles, we point out that the gist of paraconsistent logic lies on the Principle of Explosion, rather than on the Principle of Non-Contradiction, and we sharply distinguish these two also from the Principle of Non-Triviality, considering the next various weaker formulations of explosion, and investigating their interrelations.  We present then the syntactical formulations of some of the main C-systems based on classical logic, showing how several well-known logics in the literature can be recast as such a kind of C-systems, and carefully study their properties and shortcomings, showing for instance how they can be used to faithfully reproduce all classical inferences, despite being themselves only fragments of classical logic, and venturing some comments on their algebraic counterparts.  This study is intended both to fully present and characterize from scratch the field into which it inserts, as well as to set some open problems, and to point to a few directions of continuation, establishing on the way a unifying theoretical framework for further investigation for researchers involved with the foundations of paraconsistent logic.