With the advent of high-tech eras with big data, artificial intelligence and 5G communications, people have higher and higher requirements for computer performance. The traditional von Neumann architecture, whose principle is the separation of the central processing unit (CPU) and memory, not only limits the performance of the computer, but also causes a lot of energy consumption. The next generation of brain-inspired computing chips promises to break the von Neumann bottleneck by simulating the brain's neural networks, enabling a new computer architecture known as neuromorphic computing. Memristors have been found to be one of the best pieces of hardware for neuromorphic computing and the best components for building artificial neural networks. This review systematically summarizes the research progress of biomemristors as synaptic devices. Memristors and bio-synapses are first introduced, and then the research progress of biomemristors based on biomaterials and polymers is reported. Finally, the application prospects and challenges of biomemristors in synapses are pointed out. This review provides a research perspective for the application of biomemristors-based synaptic devices in artificial intelligence.