Human cognitive abilities are extraordinary. Our large brains are significantly modified from those of our closest relatives, suggesting a history of intense natural selection. The conditions favoring the evolution of human cognitive adaptations, however, remain an enigma. Hypotheses based on traditional ecological demands, such as hunting or climatic variability, have not provided satisfying explanations. Recent models based on social problem solving linked with ecological conditions offer more convincing scenarios. But it has proven difficult to identify a set of selective pressures that would have been sufficiently unique to the hominin lineage. What was so special about the evolutionary environments of our ancestors that caused them, and them alone, to diverge in such astonishing ways from their close relatives and all other life forms? Richard Alexander proposed a comprehensive integrated explanation. He argued that as our hominin ancestors became increasing able to master the traditional “hostile forces of nature,” selective pressures resulting from competition among conspecifics became increasingly important, particularly in regard to social competencies. Given the precondition of competition among kin- and reciprocity-based coalitions (shared with chimpanzees), an autocatalytic social arms race was initiated, which eventually resulted in the unusual collection of traits characteristic of the human species, such as concealed ovulation, extensive biparental care, complex sociality, and an extraordinary collection of cognitive abilities. We term this scenario the “ecological dominance–social competition” (EDSC) model and assess the feasibility of this model in light of recent developments in paleoanthropology, cognitive psychology, and neurobiology. We conclude that although strong or direct tests are difficult with current data, Alexander's model provides a far-reaching and integrative explanation for the evolution of human cognitive abilities that is consistent with evidence from a wide range of disciplines.