Phylogenetics, likelihood, evolution and complexity (PLEX) is a flexible and fast Bayesian Markov chain Monte Carlo software program for large-scale analysis of nucleotide and amino acid data using complex evolutionary models in a phylogenetic framework. The program gains large speed improvements over standard approaches by implementing partial sampling of substitution histories, a data augmentation approach that can reduce data analysis times from months to minutes on large comparative datasets. A variety of nucleotide and amino acid substitution models are currently implemented, including non-reversible and site-heterogeneous mixture models. Due to efficient algorithms that scale well with data size and model complexity, PLEX can be used to make inferences from hundreds to thousands of taxa in only minutes on a desktop computer. It also performs probabilistic ancestral sequence reconstruction. Future versions will support detection of co-evolutionary interactions between sites, probabilistic tests of convergent evolution and rigorous testing of evolutionary hypotheses in a Bayesian framework.