  Many experimental and field studies have shown that adaptation can occur very
rapidly. Two qualitatively different modes of fast adaptation have been
proposed: selective sweeps wherein large shifts in the allele frequencies occur
at a few loci and evolution via small changes in the allele frequencies at many
loci. While the first process has been thoroughly investigated within the
framework of population genetics, the latter is based on quantitative genetics
and is much less understood. Here we summarize results from our recent
theoretical studies of a quantitative genetic model of polygenic adaptation
that makes explicit reference to population genetics to bridge the gap between
the two frameworks. Our key results are that polygenic adaptation may be a
rapid process and can proceed via subtle or dramatic changes in the allele
frequency depending on the sizes of the phenotypic effects relative to a
threshold value. We also discuss how the signals of polygenic selection may be
detected in the genome. While powerful methods are available to identify
signatures of selective sweeps at loci controling quantitative traits, the
development of statistical tests for detecting small shifts of allele
frequencies at quantitative trait loci is still in its infancy.
