At a time when historical biogeography appears to be again expanding its scope after a period of focusing primarily on discerning area relationships using cladograms, new inference methods are needed to bring more kinds of data to bear on questions about the geographic history of lineages. Here we describe a likelihood framework for inferring the evolution of geographic range on phylogenies that models lineage dispersal and local extinction in a set of discrete areas as stochastic events in continuous time. Unlike existing methods for estimating ancestral areas, such as dispersal‐vicariance analysis, this approach incorporates information on the timing of both lineage divergences and the availability of connections between areas (dispersal routes). Monte Carlo methods are used to estimate branch‐specific transition probabilities for geographic ranges, enabling the likelihood of the data (observed species …