Since the beginning of the 21st century, there is no doubt that the importance of artificial intelligence has been highlighted in many fields, among which the memristor-based artificial neural network technology is expected to break through the limitation of von Neumann so as to realize the replication of the human brain by enabling strong parallel computing ability and efficient data processing and become an important way towards the next generation of artificial intelligence. A new type of nanodevice, namely memristor, which is based on the variability of its resistance value, not only has very important applications in nonvolatile information storage, but also presents obsessive progressiveness in highly integrated circuits, making it one of the most promising circuit components in the post-Moore era. In particular, memristors can effectively simulate neural synapses and build neural networks; thus, they can be applied for the preparation of various artificial intelligence systems. This study reviews the research progress of memristors in artificial neural networks in detail and highlights the structural advantages and frontier applications of neural networks based on memristors. Finally, some urgent problems and challenges in current research are summarized and corresponding solutions and future development trends are put forward.