User data traffic, especially a large amount of video traffic and small-size internet-of-things (IoT) packets, has dramatically increased in recent years with the emergence of smart devices, smart sensors and various new applications such as virtual reality and autonomous driving. It is hence crucial to increase network capacity and user access to accommodate these bandwidth consuming applications and enhance the massive connectivity. As a prominent member of the next generation multiple access (NGMA) family, non-orthogonal multiple access (NOMA) has been recognized as a promising multiple access candidate for the sixth-generation (6G) networks. The main content of this tutorial is to discuss the so-called “One Basic Principle plus Four New” concept. Starting with the basic NOMA principle to explore the possible multiple access techniques in non-orthogonal manner, the advantages and drawbacks of both the channel state information based successive interference cancelations (SIC) and quality-of-service based SIC are discussed. Then, the application of NOMA to meet the new 6G performance requirements, especially for massive connectivity, is explored. Furthermore, the integration of NOMA with new physical layer techniques is considered, followed by introducing new application scenarios for NOMA towards 6G. Finally, the application of machine learning in NOMA networks is investigated, ushering in the machine learning empowered NGMA era, for making multiple access in an intelligent manner for the next generation networks.