Solid-state memory is an essential component of the digital age. With advancements in healthcare technology and the Internet of Things (IoT), the demand for ultra-dense, ultra-low-power memory is increasing. In this review, we present a comprehensive perspective on the most notable approaches to the fabrication of physically flexible memory devices. With the future goal of replacing traditional mechanical hard disks with solid-state storage devices, a fully flexible electronic system will need two basic devices: transistors and nonvolatile memory. Transistors are used for logic operations and gating memory arrays, while nonvolatile memory (NVM) devices are required for storing information in the main memory and cache storage. Since the highest density of transistors and storage structures is manifested in memories, the focus of this review is flexible NVM. Flexible NVM components are discussed in terms of their functionality, performance metrics, and reliability aspects, all of which are critical components for NVM technology to be part of mainstream consumer electronics, IoT, and advanced healthcare devices. Finally, flexible NVMs are benchmarked and future prospects are provided.