The fifth generation (5G) wireless systems are anticipated to meet unprecedented capacity and latency requirements. In order to resolve these challenges in 5G, non-orthogonal multiple access (NOMA) is considered as a promising technique due to its ability to enhance spectrum efficiency and user access. As opposed to conventional orthogonal multiple access (OMA) which relies on orthogonal resource sharing, NOMA has a potential of supporting a higher number of users by multiplexing different users in the same resource in a non-orthogonal manner. With advanced receiver techniques, such as successive interference cancellation (SIC), the intra-user interference can be minimized at the NOMA receiver. To date, there are comprehensive surveys on NOMA, which describe the integration of NOMA with different communication technologies and discuss different NOMA classifications. However, the existing literature is scarce in reviewing state-of-the-art applications of NOMA from the perspective of its application to cellular networks (CNs), device-to-device (D2D) communications, and wireless sensor networks (WSNs). Therefore, the purpose of this survey is to fill this gap in knowledge. Specifically, NOMA with its underlying concepts are elaborated in detail. In addition, detailed system model of different NOMA-based wireless networks is presented. Furthermore, irrespective of the underlying spatial topology of the considered NOMA-based wireless network, general analytical expressions are presented to characterize the network performance. Finally, some challenges related to NOMA design are highlighted and potential research directions are pointed out to address these issues.