The end of Moore's Law and von Neumann bottleneck motivate researchers to seek alternative architectures that can fulfill the increasing demand for computation resources which cannot be easily achieved by traditional computing paradigm. As one important practice, neuromorphic computing systems (NCS) are proposed to mimic biological behaviors of neurons and synapses, and accelerate computation of neural networks. Traditional CMOS-based implementation of NCS, however, are subject to large hardware cost required to precisely replicate the biological properties. In very recent decade, emerging nonvolatile memory (eNVM) was introduced to NCS design due to its high computing efficiency and integration density. Similar to the circuits built on other nanoscale devices, eNVM-based NCS also suffers from many reliability issues. In this paper, we give a short survey about CMOS- and eNVM-based NCS, including their basic implementations and training and inference schemes in various applications. We also discuss the design challenges of these NCS and introduce some techniques that can improve the reliability, precision, scalability, and security of the NCS. At the end, we provide our insights on the design trend and future challenges of the NCS.