Emerging brain-inspired neuromorphic computing systems have become a potential candidate for overcoming the von Neuman bottleneck that limits the performance of most modern computers. Artificial synapses, used to mimic neural transmission and physical information sensing, could build highly robust and efficient computing systems similar to our brains. The employment of nanomaterials in the devices, and the device structures, are receiving a surge of interest, given the various benefits in better carrier dynamics, higher conductance, photonic interaction and photocarrier trapping, and the architectural feasibility with two and three-terminal devices. Moreover, the combination of artificial synapses and various nanomaterial-based active channels also enables visual recognition, multi-modality sensing-processing systems, hardware neural networks, etc., demonstrating appealing possibilities for practical applications. Here, we summarize the recent advances in synaptic devices based on low-dimensional nanomaterials, the novel devices with hybrid materials or structures, as well as implementation schemes of hardware neural networks. By the end of this review, we discuss the engineering issues including control methods, design complexity and fabrication process to be addressed, and envision the future developments of artificial synapse-based neuromorphic systems.