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 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 meso-scale 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.