Flexible neuromorphic electronics that emulate biological neuronal systems constitute a promising candidate for next‐generation wearable computing, soft robotics, and neuroprosthetics. For realization, with the achievement of simple synaptic behaviors in a single device, the construction of artificial synapses with various functions of sensing and responding and integrated systems to mimic complicated computing, sensing, and responding in biological systems is a prerequisite. Artificial synapses that have learning ability can perceive and react to events in the real world; these abilities expand the neuromorphic applications toward health monitoring and cybernetic devices in the future Internet of Things. To demonstrate the flexible neuromorphic systems successfully, it is essential to develop artificial synapses and nerves replicating the functionalities of the biological counterparts and satisfying the requirements for constructing the elements and the integrated systems such as flexibility, low power consumption, high‐density integration, and biocompatibility. Here, the progress of flexible neuromorphic electronics is addressed, from basic backgrounds including synaptic characteristics, device structures, and mechanisms of artificial synapses and nerves, to applications for computing, soft robotics, and neuroprosthetics. Finally, future research directions toward wearable artificial neuromorphic systems are suggested for this emerging area. Flexible neuromorphic electronics are studied for their application in next‐generation wearable computing, soft robotics, and neuroprosthetics. These applications require synaptic devices and integrated systems that are flexible, consume little power, are biocompatible, and are amenable to high‐density integration. Recent progress in flexible neuromorphic electronics, from basic background to applications is surveyed, and future research is suggested.