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A neuromorphic system is composed of hardware-based artificial neurons and synaptic devices, designed to improve the efficiency of neural computations inspired by energy-efficient and parallel operations of the biological nervous system. A synaptic device-based array can compute vector-matrix multiplication (VMM) with given input voltage signals, as a non-volatile memory device stores the weight information of the neural network in the form of conductance or capacitance. However, unlike software-based neural networks, the neuromorphic system unavoidably exhibits non-ideal characteristics that can have an adverse impact on overall system performance. In this study, the characteristics required for synaptic devices and their importance are discussed, depending on the targeted application. We categorize synaptic devices into two types: conductance-based and capacitance-based, and thoroughly explore the operations and characteristics of each device. The array structure according to the device structure and the VMM operation mechanism of each structure are analyzed, including recent advances in array-level implementation of synaptic devices. Furthermore, we reviewed studies to minimize the effect of hardware non-idealities, which degrades the performance of hardware neural networks. These studies introduce techniques in hardware and signal engineering, as well as software-hardware co-optimization, to address these non-idealities through compensation approaches. [ABSTRACT FROM AUTHOR]