Makes a sharp distinction between primacy, denoting merely a form of material power, and hegemony, understood as a legitimate practice, and as giving rise to a form of social power. Adopting an English School approach, suggests hegemony be considered as one potential institution of international society, and hence as one possible mechanism of international order. Reviews some relevant historical cases (the Concert of Europe, Pax Britannica, and Pax Americana) and argues that, instead of one model of hegemony, these represent several different variants: importantly, each displays its own distinctive legitimacy dynamics which can help us identify the possible institutional forms of hegemony in contemporary international society. This is done through three cases, examining in turn US policy on the UN Security Council, in East Asia, and on climate change. The overall argument challenges the limited post-Cold War debate about primacy, and the equally simplistic projections about the future distribution of power to which it gives rise, offering a major rethinking of the concept of hegemony in international relations.