Over the past 25 years, organic field-effect transistors (OFETs) have witnessed impressive improvements in materials performance by 3–4 orders of magnitude, and many of the key materials discoveries have been published in Advanced Materials. This includes some of the most recent demonstrations of organic field-effect transistors with performance that clearly exceeds that of benchmark amorphous silicon-based devices. In this article, state-of-the-art in OFETs are reviewed in light of requirements for demanding future applications, in particular active-matrix addressing for flexible organic light-emitting diode (OLED) displays. An overview is provided over both small molecule and conjugated polymer materials for which field-effect mobilities exceeding > 1 cm2 V–1 s–1 have been reported. Current understanding is also reviewed of their charge transport physics that allows reaching such unexpectedly high mobilities in these weakly van der Waals bonded and structurally comparatively disordered materials with a view towards understanding the potential for further improvement in performance in the future.