Neuromorphic computing is one of the most promising computing paradigms to build the next generation of energy-efficient computing systems. Many materials and devices are being investigated to develop these new architectures. Organic electrochemical transistors (OECTs) have emerged as candidate devices for the design of artificial synapses. Their ability to emulate a variety of neurological mechanisms, such as short-term and long-term plasticity, their operation relying on ionic transport and the possibility for global connectivity make them intrinsically similar to neurobiological membranes. Moreover, they present low switching energies and a wide range of tunability. Other neuromorphic components must be created with a similar technology to build an entirely organic neuromorphic system for spiking neural networks.