Non-orthogonal multiple access (NOMA) has been recently proposed as a technique to increase the network throughput and to support massive connectivity, which are major requirements in the fifth generation (5G) communication systems. The NOMA can be realized through two different approaches, namely, in (a) power-domain, and (b) code-domain. In the power-domain NOMA (PD-NOMA), multiple users are assigned different power levels – based on their individual channel quality information – over the same orthogonal resources. The functionality of PD-NOMA comprises of two main techniques, namely, superposition coding at the transmitter and successive interference cancellation (SIC) at the receiver. An efficient implementation of SIC would facilitate to remove interference across the users. The SIC is carried out at users with the best channel conditions and is performed in descending order of the channel. On the other hand, in the code-domain NOMA (CD-NOMA), multiplexing is carried out using low-density spreading sequences for each user, similar to the code division multiple access (CDMA) technology. In this article, we provide an introduction to NOMA and present the details on the working principle of NOMA systems. Later, we discuss the different types of NOMA schemes under PD- and CD-domains, and investigate the related applications in the context of 5G communication systems. Additionally, we discuss the integration of NOMA with other technologies related to 5G such as cognitive radio and massive MIMO, and discuss some future research challenges.