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In this article, Emmanuel Gaillard discusses the competing mental constructs that allow an understanding of international arbitration as a coherent phenomenon. The author identifies three such mental constructs, or representations, each of which sheds light on the entire law and practice of arbitration, and captures the underlying beliefs of the different schools of thought in the field. The three representations approach the fundamental question of the source of validity and legitimacy of the arbitral process through the prism, respectively, of the legal order of the seat of the arbitration, the different legal orders that are willing to recognize the award resulting from the arbitral process, or a distinct transnational legal order—which the author characterizes as the arbitral legal order. Each of these philosophies of international arbitration lead to distinct branding, imaging, vocabulary, and concrete conduct of all players in the field, be it practitioners, arbitrators or courts. With the study of these three representations and their consequences, as well as their foundation in cognitive sciences, Emmanuel Gaillard delivers one of the first general scientific accounts of international arbitration. [ABSTRACT FROM PUBLISHER]