In recent years, with the rapid development of integrated circuits, Moore's law has become more and more difficult to maintain. The brain-like neuromorphic computing system based on the new brain-like device synaptic transistor provides an alternative way for future computers. At present, a large number of studies have tried to fabricate synaptic transistors with different materials, and simulated synaptic behavior successfully. In this review, the basic structure and function of synaptic transistors are described, and three kinds of materials for fabricating synaptic transistors are classified: organic and non-solid electrolyte materials, inorganic solid materials and organic flexible materials, and their current research is reviewed. The purpose of this review is to provide more research directions for the research in the field by summarizing the current research on the fabrication materials of synaptic transistors. At present, there are still many new materials that have not been tried in the research field of synaptic transistors. Trying to use composite materials and two-dimensional materials is a noteworthy research direction, which is expected to promote the development of a new generation of memory technology and the application in the field of artificial intelligence.