The evolutionary history of animal cognition appears to involve a fewmajor transitions:major changes that opened up new phylogenetic possibilities for cognition. Here, we review and contrast current transitional accounts of cognitive evolution. We discuss how an important feature of an evolutionary transition should be that it changes what is evolvable, so that the possible phenotypic spaces before and after a transition are different. We develop an account of cognitive evolution that focuses on how selection might act on the computational architecture of nervous systems. Selection for operational efficiency or robustness can drive changes in computational architecture that then make new types of cognition evolvable. We propose five major transitions in the evolution of animal nervous systems. Each of these gave rise to a different type of computational architecture that changed the evolvability of a lineage and allowed the evolution of new cognitive capacities. Transitional accounts have value in that they allow a big-picture perspective of macroevolution by focusing on changes that have had major consequences. For cognitive evolution, however, we argue it is most useful to focus on evolutionary changes to the nervous system that changed what is evolvable, rather than to focus on specific cognitive capacities.