In the 5th generation (5G) of wireless communication systems, hitherto unprecedented requirements are expected to be satisfied. As one of the promising techniques of addressing these challenges, non-orthogonal multiple access (NOMA) has been actively investigated in recent years. In contrast to the family of conventional orthogonal multiple access (OMA) schemes, the key distinguishing feature of NOMA is to support a higher number of users than the number of orthogonal resource slots with the aid of non-orthogonal resource allocation. This may be realized by the sophisticated inter-user interference cancellation at the cost of an increased receiver complexity. In this article, we provide a comprehensive survey of the original birth, the most recent development, and the future research directions of NOMA. Specifically, the basic principle of NOMA will be introduced at first, with the comparison between NOMA and OMA especially from the perspective of information theory. Then, the prominent NOMA schemes are discussed by dividing them into two categories, namely, power-domain and code-domain NOMA. Their design principles and key features will be discussed in detail, and a systematic comparison of these NOMA schemes will be summarized in terms of their spectral efficiency, system performance, receiver complexity, etc. Finally, we will highlight a range of challenging open problems that should be solved for NOMA, along with corresponding opportunities and future research trends to address these challenges.