Conjugated polymers and molecular semiconductors are emerging as a viable semiconductor technology in industries such as displays, electronics, renewable energy, sensing and healthcare. A key enabling factor has been significant scientific progress in improving their charge transport properties and carrier mobilities, which has been made possible by a better understanding of the molecular structure-property relationships and the underpinning charge transport physics. Here we aim to present a coherent review of how we understand charge transport in these high-mobility van der Waals bonded semiconductors. Specific questions of interest include estimates for intrinsic limits to the carrier mobilities that might ultimately be achievable; a discussion of the coupling between charge and structural dynamics; the importance of molecular conformations and mesoscale structural features; how the transport physics of conjugated polymers and small molecule semiconductors are related; and how the incorporation of counterions in doped films-as used, for example, in bioelectronics and thermoelectric devices-affects the electronic structure and charge transport properties.