Diverse synaptic plasticity with a wide range of timescales in biological synapses plays an important role in memory, learning, and various signal processing with exceptionally low power consumption. Emulating biological synaptic functions by electric devices for neuromorphic computation has been considered as a way to overcome the traditional von Neumann architecture in which separated memory and information processing units require high power consumption for their functions. Synaptic devices are expected to conduct complex signal processing such as image classification, decision‐making, and pattern recognition in artificial neural networks. Among various materials and device architectures for synaptic devices, 2D materials and their van der Waals (vdW) heterostructures have been attracting tremendous attention from researchers based on their capacity to mimic unique synaptic plasticity for neuromorphic computing. Herein, the basic operations of biological synapses and physical properties of 2D materials are discussed, and then 2D materials and their vdW heterostructures for advanced synaptic operations with novel working mechanisms are reviewed. In particular, there is a focus on how to design synaptic devices with the vdW structures in terms of critical 2D materials and their limitations, providing insight into the emerging synaptic device systems and artificial neural networks with 2D materials.