Classical two-valued logic is very well-behaved. Sometimes it is just too much so. Several unwarranted presuppositions and expressive limitations of classical logic have motivated the introduction of many non-classical logics. Among those, paraconsistent logics are attractive in allowing for the consistency presupposition to be defeated and for a non-explosive negation to be expressed. [Some paraconsistent logics, the so-called Logics of Formal Inconsistency, can even express consistency at the object language level.] Now, if you espouse paraconsistency, it will not require from you any pledge of exclusivity. You can be fully loyal to paraconsistency while you flirt, say, with many-valuedness, or with modality. The first paraconsistent systems of deduction (Jaskowski 1948, Nelson 1959, da Costa 1963) failed both to have truth-functional semantics and to respect the replacement property, and for some time people had the impression that these failures were structural features of paraconsistency. We now know they are not. As time went by, we learned that finite-valued paraconsistent logics are feasible, and a similar thing can be said about self-extensional paraconsistent logics. My talk will reconstitute that exciting story to its latest developments. The main concentration will be on modal paraconsistency, where negated statements represent “admissible falsehoods”, as opposed to intuitionistic-like notions of “refutable truths”.