Satellite-aerial-terrestrial network (SATN) is considered as a promising architecture for sixth-generation (6G) wireless communication networks to achieve seamless coverage, flexible wireless access, and high data rate. Moreover, non-orthogonal multiple access (NOMA), and reconfigurable intelligent surface (RIS) can significantly increase spectrum and energy efficiency. Recently, the integration of these two technologies and SATN has attracted a lot of attention both in academia and industry. This survey provides a comprehensive overview of RIS-empowered SATN with NOMA. In particular, the rudimentary knowledge of SATN, NOMA scheme, and RIS technology is presented. Then, the motivations for investigating the NOMA-RIS-assisted SATN are discussed. In addition, we introduce the three usage modes of RIS, two scenarios of NOMA-RIS, and the path loss model of NOMA-RIS-assisted SATN. Next, the system performance is analyzed for a case study. Besides, a comprehensive overview of resource allocation in NOMA-RIS-assisted SATN is provided, where theoretical and artificial intelligence-based methods are compared and analyzed. Moreover, physical layer security and covert communication are selected as two representative security techniques to be discussed in NOMA-RIS-aided SATN. Furthermore, the combination of other emerging technologies with NOMA-RIS-assisted SATN is investigated. Finally, this survey provides a detailed discussion of the main challenges and open issues that need to be deeply investigated from a practical point of view, including channel modeling, channel estimation, deployment strategies, and backhaul control.