Music recommender systems (MRSs) have experienced a boom in recent years, thanks to the emergence and success of online streaming services, which nowadays make available almost all music in the world at the user’s fingertip. While today’s MRSs considerably help users to find interesting music in these huge catalogs, MRS research is still facing substantial challenges. In particular when it comes to build, incorporate, and evaluate recommendation strategies that integrate information beyond simple user–item interactions or content-based descriptors, but dig deep into the very essence of listener needs, preferences, and intentions, MRS research becomes a big endeavor and related publications quite sparse. The purpose of this trends and survey article is twofold. We first identify and shed light on what we believe are the most pressing challenges MRS research is facing, from both academic and industry perspectives. We review the state of the art toward solving these challenges and discuss its limitations. Second, we detail possible future directions and visions we contemplate for the further evolution of the field. The article should therefore serve two purposes: giving the interested reader an overview of current challenges in MRS research and providing guidance for young researchers by identifying interesting, yet under-researched, directions in the field.