Synaptic devices, including synaptic memristor and synaptic transistor, are emerging nanoelectronic devices, which are expected to subvert traditional data storage and computing methodologies. In particular, the memristive device and synaptic transistor can conduct neuromorphic computing to mimic the functions of human brain, which enables high-performance super-parallel computing, so that it overcomes the von Neumann bottleneck. Based on a new perspective and understanding, this review focuses on the discussions of synaptic devices based neuromorphic computing applications in artificial intelligence. It begins with the memristive device structure, circuit theory, fabrication method and simulation of the neuromorphic computing. Then, it focuses on the materials selection, including the 0D quantum dots, 1D nanostructure, 2D nanomaterials, 3D architectures, transition metal oxide, ferroelectric materials, alloy, and organic materials. As followed, the printable synaptic devices and typical device integration systems for neuromorphic computing applications are discussed. Finally, the future applications in neuromorphic vision, sensor, human machine intelligence, topological and quantum computing are discussed.