Many fundamental questions in evolutionary biology entail estimating rates of lineage diversification (speciation-extinction) that are modeled using birth-death branching processes. We leverage recent advances in branching-process theory to develop a flexible Bayesian framework for specifying diversification models-where rates are constant, vary continuously, or change episodically through time-and implement numerical methods to estimate parameters of these models from molecular phylogenies, even when species sampling is incomplete. We enable both statistical inference and efficient simulation under these models. We also provide robust methods for comparing the relative and absolute fit of competing branching-process models to a given tree, thereby providing rigorous tests of biological hypotheses regarding patterns and processes of lineage diversification.