The relative importance of the roles of adaptation and chance in determining genetic diversity and evolution has received attention in the last 50 years, but our understanding is still incomplete. All statements about the relative effects of evolutionary factors, especially drift, need confirmation by strong demographic observations, some of which are easier to obtain in a species like ours. Earlier quantitative studies on a variety of data have shown that the amount of genetic differentiation in living human populations indicates that the role of positive (or directional) selection is modest. We observe geographic peculiarities with some Y chromosome mutants, most probably due to a drift-related phenomenon called the surfing effect. We also compare the overall genetic diversity in Y chromosome DNA data with that of other chromosomes and their expectations under drift and natural selection, as well as the rate of fall of diversity within populations known as the serial founder effect during the recent "Out of Africa" expansion of modern humans to the whole world. All these observations are difficult to explain without accepting a major relative role for drift in the course of human expansions. The increasing role of human creativity and the fast diffusion of inventions seem to have favored cultural solutions for many of the problems encountered in the expansion. We suggest that cultural evolution has been subrogating biologic evolution in providing natural selection advantages and reducing our dependence on genetic mutations, especially in the last phase of transition from food collection to food production.