The chapter is aimed to introduce the area of memory devices based on printable organic materials. After providing an overview of current achievements, showing also some already commercialized products, various principles of data storage mechanisms are summarized together with major parameters used to characterize the memory device performance. The requirements of typical low-level target applications, such as RFID tags for smart packaging, manufacturing, and warehouse automation or security systems, are compared to the limited performance of various types of memory devices made of organic materials. OFETs with memory mechanisms based on polarization hysteresis, charging of gate dielectric or charge trapping in the active channel are discussed, including an advanced vertical OFET architecture. Particular attention is focused on nonvolatile resistive random-access memory (ReRAM) made from switchable resistive materials because of their simple two-terminal structure suitable for printing technology. The mechanisms of resistivity switching in these devices are explained, such as charge transfer, charge trapping, phase or conformation changes, or conducting filaments formation. The applicability of organic materials in new electronic elements – memristors – for unconventional memory devices used in neuromorphic computing is mentioned, too. The chapter is concluded with comments on limits of applicability of organic and polymer materials in data storage and future prospects.