The human brain is about three times as large as that of our closest living relatives, the great apes. Overall brain size is a good predictor of cognitive performance in a variety of tests in primates. Therefore, hypotheses explaining the evolution of this remarkable difference have attracted much interest. In this review, we give an overview of the current evidence from comparative studies testing these hypotheses. If cognitive benefits are diverse and ubiquitous, it is possible that most of the variation in relative brain size among extant primates is explained by variation in the ability to avoid the fitness costs of increased brain size (allocation trade‐offs and increased minimum energy needs). This is indeed what we find, suggesting that an energetic perspective helps to complement approaches to explain variation in brain size that postulate cognitive benefits. The expensive brain framework also provides a coherent scenario for how these factors may have shaped early hominin brain expansion.