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Memristors, promising nanoelectronic devices with in-memory resistive switching behavior that is assembled with a physically integrated core processing unit (CPU) and memory unit and even possesses highly possible multistate electrical behavior, could avoid the von Neumann bottleneck of traditional computing devices and show a highly efficient ability of parallel computation and high information storage. These advantages position them as potential candidates for future data-centric computing requirements and add remarkable vigor to the research of next-generation artificial intelligence (AI) systems, particularly those that involve brain-like intelligence applications. This work provides an overview of the evolution of memristor-based devices, from their initial use in creating artificial synapses and neural networks to their application in developing advanced AI systems and brain-like chips. It offers a broad perspective of the key device primitives enabling their special applications from the view of materials, nanostructure, and mechanism models. We highlight these demonstrations of memristor-based nanoelectronic devices that have potential for use in the field of brain-like AI, point out the existing challenges of memristor-based nanodevices toward brain-like chips, and propose the guiding principle and promising outlook for future device promotion and system optimization in the biomedical AI field.