  When an advantageous mutation occurs in a population, the favorable allele
may spread to the entire population in a short time, an event known as a
selective sweep. As a result, when we sample $n$ individuals from a population
and trace their ancestral lines backwards in time, many lineages may coalesce
almost instantaneously at the time of a selective sweep. We show that as the
population size goes to infinity, this process converges to a coalescent
process called a coalescent with multiple collisions. A better approximation
for finite populations can be obtained using a coalescent with simultaneous
multiple collisions. We also show how these coalescent approximations can be
used to get insight into how beneficial mutations affect the behavior of
statistics that have been used to detect departures from the usual Kingman's
coalescent.
