Wearable neuromorphic devices have gained attention because of the growth in the Internet of Things and the increasing demand for health monitoring. They provide meaningful information and interact with the external environment through physiological signal processing and seamless interaction with the human body. The concept of these devices originated from the development of neuromorphic and flexible/stretchable electronics, which offer a solution to the limitation of conventional rigid devices. They have been developed to mimic synaptic functions and flexibility/stretchability of the biological nervous system. In this study, we described the various synaptic properties that should be implemented in synaptic devices and the operating mechanisms that exhibit these properties with respect to two- and three-terminal devices. Further, we specified comprehensive methods of implementing mechanical flexibility and stretchability in neuromorphic electronics through both structure and material engineering. In addition, we explored various wearable applications of these devices, such as wearable sensors for danger detection, auxiliary equipment for people with sensory disabilities, and neuroprosthetic devices. We expect this review to provide an overall understanding of concepts and trends for flexible and stretchable neuromorphic devices, with potential extensions to state-of-the-art applications such as cybernetics and exoskeleton.