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On the basis of recent research, brain-inspired parallel computing is considered as one of the most promising technologies for efficiently handling large amounts of informational data. In general, this type of parallel computing is called neuromorphic computing; it operates on the basis of hardware-neural-network (HW-NN) platforms consisting of numerous artificial synapses and neurons. Extensive research has been conducted to implement artificial synapses with characteristics required to ensure high-level performance of HW-NNs in terms of device density, energy efficiency, and learnings accuracy. Recently, artificial synapsesspecifically, diode- and transistor-type synapsesbased on various two-dimensional (2D) van der Waals (vdW) materials have been developed. Unique properties of such 2D vdW materials allow for notable improvements in synaptic performances in terms of learning capability, scalability, and power efficiency, thereby highlighting the feasibility of the 2D vdW synapses in improving the performance of HW-NNs. In this review, we introduce the desirable characteristics of artificial synapses required to ensure high-level performance of neural networks. Recent progress in research on artificial synapses, fabricated particularly using 2D vdW materials and heterostructures, is comprehensively discussed with respect to the weight-update mechanism, synaptic characteristics, power efficiency, and scalability.