The exponential growth of information stored in data centers and computational power required for various data-intensive applications, such as deep learning and AI, call for new strategies to improve or move beyond the traditional von Neumann architecture. Recent achievements in information storage and computation in the optical domain, enabling energy-efficient, fast, and high-bandwidth data processing, show great potential for photonics to overcome the von Neumann bottleneck and reduce the energy wasted to Joule heating. Optically readable memories are fundamental in this process, and while light-based storage has traditionally (and commercially) employed free-space optics, recent developments in photonic integrated circuits (PICs) and optical nano-materials have opened the doors to new opportunities on-chip. Photonic memories have yet to rival their electronic digital counterparts in storage density; however, their inherent analog nature and ultrahigh bandwidth make them ideal for unconventional computing strategies. Here, we review emerging nanophotonic devices that possess memory capabilities by elaborating on their tunable mechanisms and evaluating them in terms of scalability and device performance. Moreover, we discuss the progress on large-scale architectures for photonic memory arrays and optical computing primarily based on memory performance.