We introduce and analyze a waiting time model for the accumulation of genetic changes. The continuous-time conjunctive Bayesian network is defined by a partially ordered set of mutations and by the rate of fixation of each mutation. The partial order encodes constraints on the order in which mutations can fixate in the population, shedding light on the mutational pathways underlying the evolutionary process. We study a censored version of the model and derive equations for an em algorithm to perform maximum likelihood estimation of the model parameters. We also show how to select the maximum likelihood partially ordered set. The model is applied to genetic data from cancer cells and from drug resistant human immunodeficiency viruses, indicating implications for diagnosis and treatment. © 2009 Biometrika Trust.