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Non-orthogonal multiple access (NOMA) is an essential enabling technology for the fifth-generation (5G) wireless networks to meet the heterogeneous demands on low latency, high reliability, massive connectivity, improved fairness, and high throughput. The key idea behind NOMA is to serve multiple users in the same resource block, such as a time slot, subcarrier, or spreading code. The NOMA principle is a general framework, and several recently proposed 5G multiple access schemes can be viewed as special cases. This survey provides an overview of the latest NOMA research and innovations as well as their applications. Thereby, the papers published in this special issue are put into the context of the existing literature. Future research challenges regarding NOMA in 5G and beyond are also discussed.