Neuromorphic computing could address the inherent limitations of conventional silicon technology in dedicated machine learning applications. Recent work on silicon-based asynchronous spiking neural networks and large crossbar arrays of two-terminal memristive devices has led to the development of promising neuromorphic systems. However, delivering a compact and efficient parallel computing technology that is capable of embedding artificial neural networks in hardware remains a significant challenge. Organic electronic materials offer an attractive option for such systems and could provide biocompatible and relatively inexpensive neuromorphic devices with low-energy switching and excellent tunability. Here, we review the development of organic neuromorphic devices. We consider different resistance-switching mechanisms, which typically rely on electrochemical doping or charge trapping, and report approaches that enhance state retention and conductance tuning. We also discuss the challenges the field faces in implementing low-power neuromorphic computing, such as device downscaling and improving device speed. Finally, we highlight early demonstrations of device integration into arrays, and consider future directions and potential applications of this technology.