We have developed a generalization of Kimura's Markov chain model for base substitution at a single nucleotide site. This generalized model incorporates more flexible transition rates and consequently allows irreversible as well as reversible chains. Because the model embodies just the right amount of symmetry, it permits explicit calculation of finite-time transition probabilities and equilibrium distributions. The model also meshes well with maximum likelihood methods for phylogenetic analysis. Quick calculation of likelihoods and their derivatives can be carried out by adapting Baum's forward and backward algorithms from the theory of hidden Markov chains. Analysis of HIV sequence data illustrates the speed of the algorithms on trees with many contemporary taxa. Analysis of some of Lake's data on the origin of the eukaryotic nucleus contrasts the reversible and irreversible versions of the model.