The conceptual structure of the new quantum theory is in some respects so different from that of classical physics that it has from the very beginning suggested radical departures in philosophy and logic. Specifically, a number of writers have considered non-standard systems of logic in connection with quantum mechanics (see bibliography). Two main directions may be discerned, initiated by Reichenbach, and by Birkhoff and von Neumann. The aim of the present paper is first to present a unified exposition of the main systems found in the literature, and second to discuss and evaluate the main logical and philosophical theses and arguments which have concerned the subject of quantum logic.