With the development of the Internet of things, artificial intelligence, and wearable devices, massive amounts of data are generated and need to be processed. High standards are required to store and analyze this information. In the face of the explosive growth of information, the memory used in data storage and processing faces great challenges. Among many types of memories, memristors have received extensive attentions due to their low energy consumption, strong tolerance, simple structure, and strong miniaturization. However, they still face many problems, especially in the application of artificial bionic synapses, which call for higher requirements in the mechanical properties of the device. The progress of integrated circuit and micro-processing manufacturing technology has greatly promoted development of the flexible memristor. The use of a flexible memristor to simulate nerve synapses will provide new methods for neural network computing and bionic sensing systems. In this paper, the materials and structure of the flexible memristor are summarized and discussed, and the latest configuration and new materials are described. In addition, this paper will focus on its application in artificial bionic synapses and discuss the challenges and development direction of flexible memristors from this perspective.