Discoveries about the cultures and cultural capacities of the great apes have played a leading role in the recognition emerging in recent decades that cultural inheritance can be a significant factor in the lives not only of humans, but of non-human animals. This prominence derives in part from the fact that these primates are those with whom we share the most recent common ancestry, thus offering clues to the origins of our own thoroughgoing reliance on cumulative cultural achievements. In addition, the intense research focus on these species has spawned an unprecedented diversity of complementary methodological approaches, the results of which suggest that cultural phenomena pervade the lives of these apes, with potentially major implications for their broader evolutionary biology. Here I review what this extremely broad array of observational and experimental methodologies has taught us about the cultural lives of chimpanzees, gorillas and orangutans, and consider the ways in which this extends our wider understanding of primate biology and the processes of adaptation and evolution that shape it. I address these issues by first evaluating the extent to which the results of cultural inheritance echo a suite of core principles that underlie organic, Darwinian evolution, but also extend them in new ways; and secondly by assessing the principal causal interactions between the primary, genetically-based organic processes of evolution, and the secondary system of cultural inheritance that is based on social learning from others.