Biologically-inspired neuromorphic computing paradigms are computational platforms that imitate synaptic and neuronal activities in the human brain to process big data flows in an efficient and cognitive manner. In the past decades, neuromorphic computing has been widely investigated in various application fields such as language translation, image recognition, modeling of phase, and speech recognition, especially in neural networks (NNs) by utilizing emerging nanotechnologies; due to their inherent miniaturization with low power cost, they can alleviate the technical barriers of neuromorphic computing by exploiting traditional silicon technology in practical applications. In this work, we review recent advances in the development of brain-inspired computing (BIC) systems with respect to the perspective of a system designer, from the device technology level and circuit level up to the architecture and system levels. In particular, we sort out the NN architecture determined by the data structures centered on big data flows in application scenarios. Finally, the interactions between the system level with the architecture level and circuit/device level are discussed. Consequently, this review can serve the future development and opportunities of the BIC system design.