The rationale for this work is to make some sort of sense of the seeming myriad of adaptive explanations for why vertebrate brains vary in size. The role that natural selection has played in brain size has been addressed using the comparative method, which allows identification of evolutionary patterns across species. One starting assumption is that brain size is a useful proxy for intelligence and therefore that large-brained animals are more intelligent than smaller-brained animals. Five classes of selection pressure form the majority of explanations: ecology, technology, innovation, sex, and sociality. After chapters in which I describe the difficulties of measuring both brain size and intelligence (cognition), I address the evidence for each of the five factors in turn, reaching the conclusion that although ecology provides the best explanations for variation in the size of brain regions, none of the factors yet offers a robust and compelling explanation for variation in whole brain size. I end by providing the steps I consider necessary to reach such an explanation, steps that I suggest are feasible, if challenging.