Robots have great application value in the fields of artificial intelligence, information technology, and intelligent manufacturing due to their efficient perception, decision-making, and execution capabilities. At present, robot learning and control have become one of the most critical frontier technologies in the field of robotics. Meanwhile, different intelligent algorithms based on neural networks have been designed to provide a planning framework for synchronous learning and control of robot systems. Specifically, the research status of neural-network-based robot learning and control is reviewed from four aspects: neural dynamics (ND) algorithms, feedforward neural networks (FNNs), recurrent neural networks (RNNs), and reinforcement learning (RL). The intelligent algorithms and related application technologies utilized for robot learning and control in the past three decades are reviewed in detail. Finally, the remaining challenges and development trends in this field are provided to promote the development of robot learning and control theory and the extension of application scenarios.