The cerebral cortex, with its conserved 6-layer structure, has inspired many unifying models of function. However, recent comparative studies of primary visual cortex have revealed considerable structural diversity, raising doubts about the possibility of an all-encompassing theory. This review examines similarities and differences in V1 across mammals. Gross laminar interconnections are relatively conserved. Major functional response classes are found universally or nearly universally. Orientation and spatial frequency tuning bandwidths are quite similar despite an enormous range of visual resolution across species, and orientation tuning is contrast-invariant. Nevertheless, there is considerable diversity in the abundance of different cell classes, laminar organization, functional architecture, and functional connectivity. Orientation-selective responses arise in different layers in different species. Some mammals have elaborate columnar architecture like orientation maps and ocular dominance bands, but others lack this organization with no apparent impact on single cell properties. Finally, local functional connectivity varies according to map structure: similar cells are connected in smooth map regions but dissimilar cells are linked in animals without maps. If there is a single structure/function relation for cortex, it must accommodate significant variations in cortical circuitry. Alternatively, natural selection may craft unique circuits that function differently in each species.