When we talk of the 'nature of intelligence', or any other attribute, we may be referring to its essential structure, or to its place in nature, particularly the function it has evolved to serve. Here I examine both, from the perspective of the evolution of intelligence in primates. Over the last 20 years, the Social (or 'Machiavellian') Intelligence Hypothesis has gained empirical support. Its core claim is that the intelligence of primates is primarily an adaptation to the special complexities of primate social life. In addition to this hypothesis about the function of intellect, a secondary claim is that the very structure of intelligence has been moulded to be 'social' in character, an idea that presents a challenge to orthodox views of intelligence as a general-purpose capacity. I shall outline the principal components of social intelligence and the environment of social complexity it engages with. This raises the question of whether domain specificity is an appropriate characterization of social intelligence and its subcomponents, like theory of mind. As a counter-argument to such specificity I consider the hypothesis that great apes exhibit a cluster of advanced cognitive abilities that rest on a shared capacity for second-order mental representation.