Can international legitimacy operate even in a deformed balance of power, and when there is only one dominant state? Conventionally, hegemony has been perceived as a threat to international society. But how then is international order to be maintained, if this still requires a managerial role on the part of the great powers? IR theory has not taken that problem sufficiently seriously. This study 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, the author suggests hegemony be considered as one potential institution of international society, and hence as one possible mechanism of international order. The book 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. Once these are appreciated, they 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. In doing so, it offers a major re-thinking of the concept of hegemony in international relations.