In the era of information explosion, silicon‐based storage can barely meet the burgeoning demand for data storage and making Moore's law no longer accurate. What is more, the traditional von Neumann architecture separates the computing module from the storage unit, which causes a slowing down of the overall computing efficiency. Conjugated polymer memristors, due to the advantages of simple fabrication, convenient operation and low cost, are considered as promising candidates for next‐generation memory and neuromorphic computing applications. This review summarizes the advances in conjugated polymer memristive materials and devices made during the first two decades of the 21st century, and their potential applications in information storage and neuromorphic computing. Focuses are on switching mechanisms and materials classifications, as well as memory, artificial synapse and bioinspired and in‐memory computing performance. © 2020 Society of Chemical Industry