Resistive memory devices based on organic materials that can be configured to two or more stable resistance states have been extensively explored as information storage media due to their advantages, which include simple device structures, low fabrication costs, and flexibility. Various organic-based materials such as small molecules, polymers, and composite materials have been observed to show bistability. This review provides a general summary about the materials, structures, characteristics, and mechanisms of organic resistive memory devices. Several critical strategies for device fabrication, performance enhancement, and integrated circuit architectures are also discussed.