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Nonvolatile memory (NVM)‐based neuromorphic computing has been attracting considerable attention from academia and the industry. Although it is not completely successful yet, remarkable achievements have been reported pertaining to synaptic devices that can leverage NVM capable of storing multiple states. The analog synaptic devices performing computation similar to biological nerve systems are crucial in energy‐efficient analog neuromorphic computing systems. To use NVM as an analog synaptic device, researchers focus on improving device characteristics. Among various characteristics, the most challenging one is linearity and symmetry of synaptic weight update that is required for on‐chip training. In this regard, this review paper discusses recent synaptic device improvements focusing on novel schemes tailored for each NVM device to improve the linearity and symmetry. In addition to device‐level studies, recent research achievements are reviewed expanded up to chip‐level studies because in realizing neuromorphic hardware systems beyond a single synaptic device, several considerations and requirements are needed to confirm for high‐level design, and accordingly, cooptimize among synaptic devices, synapse arrays, electrical circuits, neural networks, algorithms, and implementation. Also, this review introduces various circuit and algorithmic approaches to compensate for the non‐ideality of the analog synaptic device. [ABSTRACT FROM AUTHOR]