Abstract: The efficient, fast and accurate information transmission in the large-scale neural network in the brain is the exact origin that the brain can control the complex life activities of humans and animals and enable organisms to survive in the changeable natural environment. As an important medium for information transmission between neurons, the synapses ensure the efficient operation of neural networks. Therefore, to build electronic synapses with synaptic functions is one essential way to study bionic systems and brain-like neural networks. Researchers have previously tried to simulate synaptic functions with various electronic devices, among which memristor has become one good candidate to build neuromorphic synapses due to its unique device structure and memory characteristics. The researches of memristor-based synapses in recent years are comprehensively summarized in this article, including the synaptic plasticity, metaplasticity, non-associative learning, associative learning and other functions. It also summarizes the application, problems and challenges in artificial neural networks, as well as the research prospects of memristor-based synapses.