One of Waltz’s major contributions was the idea that political structures can be specified, in a rough first approximation, by ordering principle, functional differentiation, and distribution of capabilities – an understanding that remains largely taken-for-granted in contemporary IR. This chapter shows, however, that this tripartite conception can neither accurately nor fruitfully depict the structure of three simple anarchic systems: the Hobbesian state of nature, immediate-return forager societies, and great power states systems. In fact, Waltz’s depiction of great power states systems, his implicit model of a generic international system, is wildly inaccurate on all three of his dimensions of structure. Great power states systems, rather than lack hierarchy, are structured by the hierarchical superiority of states and great powers. Great powers, states, and nonstate actors perform different political functions. And the standard Waltzian account of the distribution of capabilities as a matter of the number of great powers (“polarity”) is about as useful as depicting the distribution of wealth in a society by the number of billionaires.