Many organic electronic devices exhibit switching behavior, and have therefore been proposed as the basis for a nonvolatile memory (NVM) technology. This Review summarizes the materials that have been used in switching devices, and describes the variety of device behavior observed in their charge–voltage (capacitive) or current–voltage (resistive) response. A critical summary of the proposed charge‐transport mechanisms for resistive switching is given, focusing particularly on the role of filamentary conduction and of deliberately introduced or accidental nanoparticles. The reported device parameters (on–off ratio, on‐state current, switching time, retention time, cycling endurance, and rectification) are compared with those that would be necessary for a viable memory technology.