To learn about the past from a sample of genomic sequences, one needs to
understand how evolutionary processes shape genetic diversity. Most population
genetic inference is based on frameworks assuming adaptive evolution is rare.
But if positive selection operates on many loci simultaneously, as has recently
been suggested for many species including animals such as flies, a different
approach is necessary. In this review, I discuss recent progress in
characterizing and understanding evolution in rapidly adapting populations
where random associations of mutations with genetic backgrounds of different
fitness, i.e., genetic draft, dominate over genetic drift. As a result, neutral
genetic diversity depends weakly on population size, but strongly on the rate
of adaptation or more generally the variance in fitness. Coalescent processes
with multiple mergers, rather than Kingman's coalescent, are appropriate
genealogical models for rapidly adapting populations with important
implications for population genetic inference.