Neuromorphic computing can process large amounts of information in parallel and provides a powerful tool to solve the von Neumann bottleneck. Constructing an artificial neural network (ANN) is a common means to realize neuromorphic computing, which has exhibited potential applications in pattern recognition, complex sensing, and other areas. Reservoir computing (RC), which is another approach to realize neuromorphic computing, has shown some progress and attracted researchers' attention. Neuromorphic computing can be generally implemented by fabricating memristive array systems. 2D-material-based memristive systems and their applications in ANN and RC have been investigated substantially in recent years due to the unique properties of these systems, such as atomic-level thickness and high carrier mobility. In this Review, we first discuss the volatility and nonvolatility properties of memristive devices and their applications in ANN and RC. Second, 2D materials that can be used to fabricate these devices are introduced, and their classification, physical properties, and preparation methods are presented. Third, we discuss the working mechanisms of 2D-material-based synaptic devices, the mimicked synaptic functions, and the applications of these devices in neuromorphic computing through ANN and RC. Lastly, the performance, progress, and future development directions of 2D-material-based synaptic devices are analyzed. This work systematically investigates the status of 2D-material-based synaptic devices and promotes their utilization in neuromorphic computing.