Living organisms have a long evolutionary history that has provided them with functions and structures that enable them to survive in their environment. The goal of biomimetic technology is to emulate these traits of living things. Research in bioinspired electronics develops electronic sensors and motor systems that mimic biological sensory organs and motor systems and that are intended to be used in bioinspired applications such as humanoid robots, exoskeletons, and other devices that combine a living body and an electronic device. To develop bioinspired robotic and electronic devices that are compatible with the living body at the neuronal level and that are operated by mechanisms similar to those in a living body, researchers must develop biomimetic electronic sensors, motor systems, brains, and nerves. Artificial organic synapses have emulated the brain's plasticity with much simpler structures and lower fabrication cost than neurons based on silicon circuits, and with smaller energy consumption than traditional von Neumann computing methods. Organic synapses are promising components of future neuromorphic systems. In this Account, we review recent research trends of neuromorphic systems based on organic synapses, then suggest research directions. We introduce the device structures and working mechanisms of reported organic synapses and the brain's plasticity, which are mainly imitated to demonstrate the learning and memory function of the organic synapses. We also introduce recent reports on sensory synapses and sensorimotor nervetronics that mimic biological sensory and motor nervous systems. Sensory nervetronics can be used to augment the sensory functions of the living body and to comprise the sensory systems of biomimetic robots. Organic synapses can also be used to control biological muscles and artificial muscles that have the same working mechanism as biological muscle. Motor nervetronics would impart life-like motion to bioinspired robots. Chemical approaches may provide insights to guide development of new organic materials, device structures, and working mechanisms to improve synaptic responses of organic neuromorphic systems. For example, organic synapses can be applied to electronic and robotic skins and bioimplantable medical devices that use mechanically stable, self-healing, and biocompatible organic materials. Biochemical approaches may expand the plasticity of the brain and nervous system. We expect that organic neuromorphic systems will be vital components in bioinspired robotic and electronic applications, including biocompatible neural prosthetics, exoskeletons, humanoid soft robots, and cybernetics devices that are integrated with biological and artificial organs.