This chapter explores the potential of cryogenic computing to address the limitations of CMOS technology in handling modern data-intensive applications. Cryogenic environments, particularly those utilizing superconducting devices, offer substantial reductions in electrical resistance, leading to significant improvements in speed and energy efficiency. Such advancements are critical for quantum computing, high-performance computing, and space exploration. However, the lack of scalable, efficient cryogenic memory systems poses a significant hurdle. The chapter introduces the concept of cryogenic memory and provides a comprehensive review of existing cryogenic non-superconducting technologies, including charge-based SRAM, DRAM, Flash, and resistance-based memristive, spintronic, ferroelectric, phase-change, and topological memories. It also examines hybrid memory systems that integrate non-superconducting storage with superconducting access technologies. The analysis highlights the performance enhancements achieved at cryogenic temperatures while addressing the current challenges associated with these technologies. By scrutinizing these advancements and obstacles, this chapter outlines ongoing research efforts and future directions essential for the development of effective cryogenic memory technologies, paving the way for the next generation of high-performance, energy-efficient computing systems.