With the rapid development of wearable devices, flexible conductive materials, which are one of the most important components of flexible electronics, have continued to attract increasing attention as important materials. Conventional electrodes mainly consist of rigid metallic materials, and consequently lack flexibility. Some of the strategies commonly used to make flexible metal electrodes include reducing the thickness of the electrode and designing electrodes with unique structural features. However, these techniques are generally complicated and expensive. Nanocarbon materials, especially carbon nanotubes and graphene, are highly flexible and exhibit excellent conductivity, superior thermal stability, good chemical stability, and high transmittance, making them good alternative materials for the preparation of flexible conductors. In this review, we have summarized recent advances towards the development of flexible conductors based on different types of nanocarbon materials, including carbon nanotubes arrays, carbon nanotubes films, carbon nanotubes fibers, graphene prepared using exfoliation or chemical vapor deposition techniques and graphene fibers. We have also provided a brief review of flexible conductive materials based on graphene/carbon nanotube composites, as well as a summary of the synthesis, fabrication and performances of these conductors. Finally, we have discussed the future challenges and possible research directions of flexible conductors based on nanocarbon materials.