This paper will review emerging non-volatile memory (NVM) technologies, with the focus on phase change memory (PCM), spin-transfer-torque random-access-memory (STTRAM), resistive random-access-memory (RRAM), and ferroelectric field-effect-transistor (FeFET) memory. These promising NVM devices are evaluated in terms of their advantages, challenges, and applications. Their performance is compared based on reported parameters of major industrial test chips. Memory selector devices and cell structures are discussed. Changing market trends toward low power (e.g., mobile, IoT) and data-centric applications create opportunities for emerging NVMs. High-performance and low-cost emerging NVMs may simplify memory hierarchy, introduce non-volatility in logic gates and circuits, reduce system power, and enable novel architectures. Storage-class memory (SCM) based on high-density NVMs could fill the performance and density gap between memory and storage. Some unique characteristics of emerging NVMs can be utilized for novel applications beyond the memory space, e.g., neuromorphic computing, hardware security, etc. In the beyond-CMOS era, emerging NVMs have the potential to fulfill more important functions and enable more efficient, intelligent, and secure computing systems.