We introduce a general class of models for sequence evolution that includes network phylogenies. Networks, a generalization of strictly tree-like phylogenies, are proposed to model situations where multiple lineages contribute to the observed sequences. An algorithm to compute the probability distribution of binary character-state configurations is presented and statistical inference for this model is developed in a likelihood framework. A stepwise procedure based on likelihood ratios is used to explore the space of models. Starting with a star phylogeny, new splits (nontrivial bipartitions of the sequence set) are successively added to the model until no significant change in the likelihood is observed. A novel feature of our approach is that the new splits are not necessarily constrained to be consistent with a treelike mode of evolution. The fraction of invariable sites is estimated by maximum likelihood simultaneously with other model parameters and is essential to obtain a good fit to the data. The effect of finite sequence length on the inference methods is discussed. Finally, we provide an illustrative example using aligned VP1 genes from the foot and mouth disease viruses (FMDV). The different serotypes of the FMDV exhibit a range of treelike and network evolutionary relationships.