Following the trend of miniaturization as per Moore's law, and facing the strong demand of next-generation electronic devices that should be highly portable, wearable, transplantable, and lightweight, growing endeavors have been made to develop novel flexible data storage devices possessing nonvolatile ability, high-density storage, high-switching speed, and reliable endurance properties. Nonvolatile organic data storage devices including memory devices on the basis of floating-gate, charge-trapping, and ferroelectric architectures, as well as organic resistive memory are believed to be favorable candidates for future data storage applications. In this Review, typical information on device structure, memory characteristics, device operation mechanisms, mechanical properties, challenges, and recent progress of the above categories of flexible data storage devices based on organic nanoscaled materials is summarized.