Although silicon technology has been the main driving force for miniaturizing device dimensions to improve cost and performance, the current application of Si to soft electronics (flexible and stretchable electronics) is limited due to material rigidity. As a result, various prospective materials have been proposed to overcome the rigidity of conventional Si technology. In particular, nano-carbon materials such as carbon nanotubes (CNTs) and graphene are promising due to outstanding elastic properties as well as an excellent combination of electronic, optoelectronic, and thermal properties compared to conventional rigid silicon. The uniqueness of these nano-carbon materials has opened new possibilities for soft electronics, which is another technological trend in the market. This review covers the recent progress of soft electronics research based on CNTs and graphene. We discuss the strategies for soft electronics with nano-carbon materials and their preparation methods (growth and transfer techniques) to devices as well as the electrical characteristics of transparent conducting films (transparency and sheet resistance) and device performances in field effect transistor (FET) (structure, carrier type, on/off ratio, and mobility). In addition to discussing state of the art performance metrics, we also attempt to clarify trade-off issues and methods to control the trade-off on/off versus mobility). We further demonstrate accomplishments of the CNT network in flexible integrated circuits on plastic substrates that have attractive characteristics. A future research direction is also proposed to overcome current technological obstacles necessary to realize commercially feasible soft electronics.