Do more complex brains operate on the same principles as simpler brains, merely employing more of the same; or has evolution produced new principles? A neglected research agenda is the search for relevant differences between brains of animals belonging to different major grades of complexity and cognitive capacity. More complex brains are believed to be capable of more transactions, discriminations, memory and repertoire - functional criteria of 'better' brains. While comparative cognition needs to test these expectations to verify what would be one of the major consequences of evolution, comparative neurobiology needs to discover what, in detail, by all the methods and measures of neuroanatomy, neurophysiology and neurochemistry, is different in more complex brains and which of these differences are relevant to behavioral differences. Formerly it was fashionable to deny differences in the dimension that might be called higher and lower. The criterion of complexity, defined as the number of different parts, processes, interactions and behaviors, may be a useful index of advancement. Unequivocal advancement has occurred between some major taxa, although it is not inevitable. Anatomically, it is clear that novelties have evolved and complexity is more than size or number of the same components. Physiologically, new types of cells, local and larger circuits and emergent properties of assemblies have increased the complexity of operations and organization. It needs to be reasserted that more complex brains have evolved repeatedly, both among invertebrates and among vertebrates, because similar assertions have been incorrectly labeled as 'intuitive scaling' and anthropocentric or moral statements. None of these or any necessary implications about tendencies or causation need be involved. The main thrust of this essay is to underline three propositions as steps toward a plan of action: (1) We are deeply ignorant of the exact ways in which the more and the less complex brains differ - emphasizing those from higher taxonomic categories clearly different in grades of complexity. (2) The needed comparisons require the methods of natural history prior to mechanistic analysis, since unfamiliar traits must be uncovered. (3) Especially to include the higher nervous functions in these comparisons, new approaches are needed. I perceive a major frontier of new insights about brains. Pursuit of this large agenda of research will be profoundly significant both for neurobiology and general biology.