Many of the current artificial intelligence (AI) applications that are rapidly becoming indispensable in our society rely on software-based artificial neural networks or deep learning algorithms that are powerful, but energy-inefficient. The brain in comparison is highly efficient at similar classification and pattern finding tasks. Neuromorphic engineering attempts to take advantage of the efficiency of the brain by mimicking several crucial concepts to efficiently emulate AI tasks. Organic electronic materials have been particularly successful in mimicking both the basic functionality of the brain, including important spiking phenomena, but also in low-power operation of hardware-implemented artificial neural networks as well as interfacing with physiological environments due to their biocompatible nature. This article provides an overview of the basic functional operation of the brain and its artificial counterparts, with a particular focus on organic materials and devices. We highlight efforts to mimic brain functions such as spatiotemporal processing, homeostasis, and functional connectivity and emphasize current challenges for efficient neuromorphic computing applications. Finally, we present our view of future directions in this exciting and rapidly growing field of organic neuromorphic devices.