Robots are widely used, providing significant convenience in daily life and production. With the rapid development of artificial intelligence and neuromorphic computing in recent years, the realization of more intelligent robots through a profound intersection of neuroscience and robotics has received much attention. Neuromorphic circuits based on memristors used to construct hardware neural networks have proved to be a promising solution of shattering traditional control limitations in the field of robot control, showcasing characteristics that enhance robot intelligence, speed, and energy efficiency. Starting with introducing the working mechanism of memristors and peripheral circuit design, this review gives a comprehensive analysis on the biomimetic information processing and biomimetic driving operations achieved through the utilization of neuromorphic circuits in brain-like control. Four hardware neural network approaches, including digital-analog hybrid circuit design, novel device structure design, multi-regulation mechanism, and crossbar array, are summarized, which can well simulate the motor decision-making mechanism, multi-information integration and parallel control of brain at the hardware level. It will be definitely conductive to promote the application of memristor-based neuromorphic circuits in areas such as intelligent robotics, artificial intelligence, and neural computing. Finally, a conclusion and future prospects are discussed.