Neuromorphic photonics system based on the principle of biological brain is emerging as one of the potential solutions to the bottleneck inherent in classical von Neumann computing system. Optoelectronic synaptic devices, used to mimic the visual function of bio-synapse by adapting synaptic weights, can construct a highly efficient brain-inspired computing system, in which the nanostructured materials and device architectures are attracting extensive interests, giving many potential benefits in confined light-matter interaction, fast carrier dynamics, and photocarriers trapping. Moreover, the conjunction of traditional nanostructured photodetectors and newly realized electronic synapses also exhibit appealing performance for many practical applications including information processing and computing. This review focuses and summarizes on recent achievement in developing advanced optoelectronic synaptic devices based on nanostructured materials as 0D (quantum dots), 1D, and 2D materials, as well as, the rapidly evolving hybrid heterostructures. In addition, challenges and promising prospects in this research field are also discussed.