Massive multiple-input multiple-output (MIMO) and non-orthogonal multiple access (NOMA) are two key techniques for enabling massive connectivity in future wireless networks. A massive MIMO-NOMA system can deliver remarkable spectral improvements and low communication latency. Nevertheless, the uncontrollable stochastic behavior of the wireless channels can still degrade its performance. In this context, the idea of an intelligent reflecting surface (IRS) has emerged as a promising technology for smartly overcoming the possibly detrimental effects of the wireless environment. The disruptive IRS concept of controlling the propagation channels via software can provide attractive performance gains to the communication networks, including higher data rates, improved user fairness, and possibly higher energy efficiency. In this article, we demonstrate the main roles of IRSs in MIMO-NOMA systems. Specifically, we identify key challenges and perform a comprehensive discussion of the main performance gains that can be achieved in IRS-assisted massive MIMO-NOMA (IRS-NOMA) networks. We outline exciting futuristic use case scenarios for IRS-NOMA and expose the main related challenges and future research directions. Furthermore, throughout the article, we support our in-depth discussions with representative numerical results.