With the rapid development of deep learning, the current rapid update and iteration of intelligent algorithms put forward high requirements for hardware computing power. Limited by the exhaustion of Moore’s law and the von Neumann bottleneck, the traditional CMOS integration cannot meet the urgent needs of hardware computing power improvement. The utilization of new device memristors to construct a neuromorphic computing system can realize the integration of storage and computing, and has the characteristics of extremely high parallelism and ultra-low power consumption. In this work, the device structure and physical mechanism of mainstream memristors are reviewed in bottom-to-top order firstly, and their performance characteristics are compared and analyzed. Then, the recent research progress of memristors to realize artificial neurons and artificial synapses is introduced, including the simulation of specific circuit forms and neuromorphic functions. Secondly, in this work, the structural forms of passive and active memristive arrays and their applications in neuromorphic computing, including neural network-based handwritten digits and face recognition, are reviewed. Lastly, the current challenges of memristive brain-like computing from the bottom to the top, are summarized and the future development of this field is also prospected.