Our narrative literature review acknowledges that, although there is an increasing interest in recommender system fairness in general, the music domain has received relatively little attention in this regard. However, addressing fairness of music recommender systems (MRSs) is highly important because the performance of these systems considerably impacts both the users of music streaming platforms and the artists providing music to those platforms. The distinct needs that these stakeholder groups may have, and the different aspects of fairness that therefore should be considered, make for a challenging research field with ample opportunities for improvement. The review first outlines current literature on MRS fairness from the perspective of each stakeholder and the stakeholders combined, and then identifies promising directions for future research. The two open questions arising from the review are as follows: (i) In the MRS field, only limited data is publicly available to conduct fairness research; most datasets either originate from the same source or are proprietary (and, thus, not widely accessible). How can we address this limited data availability? (ii) Overall, the review shows that the large majority of works analyze the current situation of MRS fairness, whereas only few works propose approaches to improve it. How can we move forward to a focus on improving fairness aspects in these recommender systems? At FAccTRec '22, we emphasize the specifics of addressing RS fairness in the music domain.