Over recent years there has been tremendous progress in developing low-temperature, solution-processible organic semiconductors that provide high charge carrier mobilities for both n-type and p-type device operation, good operational stability and other functionalities such as efficient electroluminescene, sensing or memory functions for a variety of applications. Here we are interested in understanding the charge transport physics of high mobility conjugated polymers and the relationship between molecular structure, polymer microstructure and charge transport. Many of the recently discovered high mobility polymers, in particular donor-acceptor copolymers, are characterised by a puzzling lack of pronounced crystalline order. In this presentation we will present our current understanding of the transport physics of these materials and of the reasons why these materials exhibit such high carrier mobilities.