In the wake of growth in intelligent mobile devices and wide usage of bandwidth-hungry applications of mobile Internet, the demand of wireless data traffic and ubiquitous mobile broadband is rapidly increasing. On account of these developments, the research on fifth generation (5G) networks presents an accelerative tendency on a global scale. Edge computing draw lots of attention for reducing the time delay and improving the Quality of Service for the networks. While, fog radio access networks (F-RANs) is an emergent architecture, which takes full use of edge computing and distributed storing capabilities in edge devices. In this article, we propose an architecture of non-orthogonal multiple access (NOMA) based F-RANs, which has a strong capability of edge computing and can meet the heterogeneous requirements in 5G systems. NOMA with successive interference cancellation (SIC) is regarded as a critical multi-user access technology. In NOMA, more than one user can access the same time, code domain, and frequency resources. With assigning different power levels to multi-user and implementing SIC, multiple users detection can be achieved. In this article, we provide a description of the NOMA based F-RANs architecture, and discuss the resource allocation in that. We will focus on the power and subchannel allocation in consideration of using NOMA and the edge caching. Simulation results show that the proposed NOMA baesd F-RANs architecture and the resource management mechanisms can achieve the high net utility for the RANs.