Field-effect transistors based on conjugated polymers are being developed for large-area electronic applications on flexible substrates, but they also provide a very useful tool to probe the charge transport physics of these complex materials. In this review we discuss recent progress in polymer semiconductor materials, which have brought the performance and mobility of polymer devices to levels comparable to that of small-molecule organic semiconductors. These new materials have also enabled deeper insight into the charge transport physics of high-mobility polymer semiconductors gained from experiments with high charge carrier concentration and better molecular-scale understanding of the electronic structure at the semiconductor/dielectric interface.