Driven by the rapid escalation of the wireless capacity requirements imposed by advanced multimedia applications (e.g., ultra-high-definition video, virtual reality etc.), as well as the dramatically increasing demand for user access required for the Internet of Things (IoT), the fifth generation (5G) networks face challenges in terms of supporting large-scale heterogeneous data traffic. Non-orthogonal multiple access (NOMA), which has been recently proposed for the 3rd generation partnership projects long-term evolution advanced (3GPP-LTE-A), constitutes a promising technology of addressing the above-mentioned challenges in 5G networks by accommodating several users within the same orthogonal resource block. By doing so, significant bandwidth efficiency enhancement can be attained over conventional orthogonal multiple access (OMA) techniques. This motivated numerous researchers to dedicate substantial research contributions to this field. In this context, we provide a comprehensive overview of the state-of-the-art in power-domain multiplexing aided NOMA, with a focus on the theoretical NOMA principles, multiple antenna aided NOMA design, on the interplay between NOMA and cooperative transmission, on the resource control of NOMA, on the co-existence of NOMA with other emerging potential 5G techniques and on the comparison with other NOMA variants. We highlight the main advantages of power-domain multiplexing NOMA compared to other existing NOMA techniques. We summarize the challenges of existing research contributions of NOMA and provide potential solutions. Finally, we offer some design guidelines for NOMA systems and identify promising research opportunities for the future.