Human evolution is interesting from a number of reasons. Among them is the fact that human ancestors lived in Africa for millions of years, but they successfully colonized non-African environments only in the last 50,000 years or so. Many of these areas differ substantially in ecological terms, and their colonization was accompanied by selective pressures to adapt locally. Analyzing modern and ancient genomes we infer that local adaptation significantly contributed to the (otherwise modest) genic differences that exist among human groups. Some of these adaptations result in population differences in important phenotypes, including disease. The origin of the adaptive alleles remains debated, with new, introgressed, and neutral standing alleles being plausible sources. I will discuss how shifts in natural selection can mediate fast novel adaptations, focusing on how long-term balancing selection (an important type of natural selection that maintains advantageous diversity within populations) served as a source of alleles that mediated recent Eurasian adaptations. When applied to data in primates, these approaches help us understand also the evolution of our closest living relatives.