Contrary to orthogonal multiple-access (OMA), non-orthogonal multiple-access (NOMA) schemes can serve a pool of users without exploiting the scarce frequency or time domain resources. This is useful in meeting the future network requirements (5G and beyond systems), such as, low latency, massive connectivity, users' fairness, and high spectral efficiency. On the other hand, content caching restricts duplicate data transmission by storing popular contents in advance at the network edge which reduces data traffic. In this survey, we focus on cache-aided NOMA-based wireless networks which can reap the benefits of both cache and NOMA; switching to NOMA from OMA enables cache-aided networks to push additional files to content servers in parallel and improve the cache hit probability. Beginning with fundamentals of the cache-aided NOMA technology, we summarize the performance goals of cache-aided NOMA systems, present the associated design challenges, and categorize the recent related literature based on their application verticals. Concomitant standardization activities and open research challenges are highlighted as well.