Recent analyses of the fossil record and molecular phylogenies suggest that there are fundamental limits to biodiversity, possibly arising from constraints in the availability of space, resources, or ecological niches. Under this hypothesis, speciation rates decay over time and biodiversity eventually saturates, with new species emerging only when others are driven to extinction. This view of macro-evolution contradicts an alternative hypothesis that biodiversity is unbounded, with species ever accumulating as they find new niches to occupy. These contrasting theories of biodiversity dynamics yield fundamentally different explanations for the disparity in species richness across taxa and regions. Here, we test whether speciation rates have decayed or remained constant over time, and whether biodiversity is saturated or still expanding. We first derive a general likelihood expression for internode distances in a phylogeny, based on the well-known coalescent process from population genetics. This expression accounts for either time-constant or time-variable rates, time-constant or time-variable diversity, and completely or incompletely sampled phylogenies. We then compare the performance of different diversification scenarios in explaining a set of 289 phylogenies representing amphibians, arthropods, birds, mammals, mollusks, and flowering plants. Our results indicate that speciation rates typically decay over time, but that diversity is still expanding at present. The evidence for expanding-diversity models suggests that an upper limit to biodiversity has not yet been reached, or that no such limit exists. © 2010 Morlon et al.