In the exciting era of artificial intelligence (AI) and big data, new advanced data storage and processing technologies are of great interest. The novel neuromorphic computing hardware based on nanoscale memristive materials holds the potential to overcome the von Neumann bottleneck. Specifically, carbon nanomaterials have emerged as a promising tool for the next generation of neuromorphic devices. This paper comprehensively reviews the recent advances in memristors based on 0D, 1D, and 2D carbon nanostructure materials for neuromorphic applications.