The propagation environment was uncontrollable in first-generation to fifth-generation (5G) wireless technologies. This behavior of the wireless propagation environment is one of the prime constraints in harnessing the performance of wireless networks. This problem can be addressed in sixth-generation (6G) wireless networks by deploying intelligent reflecting surfaces (IRSs). IRS’s amplitude and phase reflecting coefficient of reflecting units (RUs) can be adjusted via a programmable controller to meet the network requirements. On the other hand, in 5G and 6G wireless communication networks, non-orthogonal multiple access (NOMA) is a robust and well-admired multiple access scheme among the other multiple access counterparts in terms of spectrum efficiency and link capacity. NOMA allows many user equipment (UE) by utilizing non-orthogonal distribution of resources. Therefore, the combination of IRS and NOMA is one of the dominant technologies for 6G wireless networks. Based upon the importance of NOMA and IRS in the initial development of 6G wireless networks, this paper presents a comprehensive survey on IRS-assisted NOMA-based networks, considering their designs and challenges. In this work, the concept and structure of IRS-assisted NOMA have been explained with an in-depth analysis of the frameworks. It also includes some challenges of IRS-assisted NOMA in wireless communication networks. Further, applications and future research directions of IRS-assisted NOMA networks are discussed.