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The rapid development of artificial intelligence (AI), big data analytics, cloud computing, and Internet of Things applications expect the emerging memristor devices and their hardware systems to solve massive data calculation with low power consumption and small chip area. This paper provides an overview of memristor device characteristics, models, synapse circuits, and neural network applications, especially for artificial neural networks and spiking neural networks. It also provides research summaries, comparisons, limitations, challenges, and future work opportunities.