The explosive growth of mobile devices and the rapid increase of wideband wireless services call for advanced communication techniques that can achieve high spectral efficiency and meet the massive connectivity requirement. Cognitive radio (CR) and non-orthogonal multiple access (NOMA) are envisioned to be important solutions for the fifth generation wireless networks. Integrating NOMA techniques into CR networks (CRNs) has the tremendous potential to improve spectral efficiency and increase the system capacity. However, there are many technical challenges due to the severe interference caused by using NOMA. Many efforts have been made to facilitate the application of NOMA into CRNs and to investigate the performance of CRNs with NOMA. This article aims to survey the latest research results along this direction. A taxonomy is devised to categorize the literature based on operation paradigms, enabling techniques, design objectives and optimization characteristics. Moreover, the key challenges are outlined to provide guidelines for the domain researchers and designers to realize CRNs with NOMA. Finally, the open issues are discussed.