Coalescent theory has many new branches.

Coalescent theory and the story of its unfolding have become part of the canon of population genetics. This is not a signal that the field is in decline, but amark of the enduring value of the genegenealogical way of thinking. Within both the biological and the mathematical literature of coalescent theory, novel extensions and wholly new developments continue to appear. This special issue of the journal presents a snapshot of coalescent theory’s leading edge, now roughly thirty years after the birth the field. Coalescent theory begins by imagining the ancestry of a sample of size n at a single genetic locus without recombination. This ancestry is comprised of exactly n − 1 coalescent events wherein pairs of genetic lineages join together backward in time. The result is called the gene genealogy of the sample. At the (n − 1)th coalescent event, the most recent common ancestor of the entire sample has been reached, and the process is stopped because the fundamental aim of coalescent theory is to understand genetic variation within samples. All genetic variation in a sample must be the result ofmutations that occurred on these branches of the gene genealogy, between the present and the time of the most recent common ancestor. Kingman (1982a,c,b) described his n-coalescent using the discrete-time, haploid, exchangeable population model of Cannings (1974), with its now familiar distribution of ‘‘offspring numbers’’. In a population of size N , if νi denotes the number of offspring of individual i in some generation, then the overall outcome of reproduction in that generation is the vector (ν1, . . . , νN). These offspring numbers are exchangeable random variables in that they are identically distributed and have only a mild sort of non-independence, specifically that N i=1 νi = N . Making the further assumptions thatN is constant over time and that the genotypes of individuals do not affect the distribution of offspring numbers, we have the basic ingredients of Kingman’s model and many of its later extensions. From a more biological point of view, in addition to this explicit statement of constant population size,wemust recognize a number of other assumptions. First, the organisms are haploid. The oft-stated view that the coalescent holds for diploid populations if N is replaced by 2N comes from the analysis of the diploid, monecious Wright–Fisher model (Fisher, 1930; Wright, 1931) which is, in essence, a haploidmodel. Second, there is no selection; all genetic variation is assumed to be neutral. Third, there is no geographic structure. For haploids, this means that the locations of individuals (and resulting population densities, etc.) have no effect on their numbers of offspring. The same restriction applies to diploids, but for diploids it must also be true that geographic distance poses no barrier to mating. All three caveats are aspects of the fundamental assumption behind exchangeability—namely