Future cellular networks will increasingly rely on the millimeter-wave bands to increase capacity. Migrating to ever higher carrier frequencies will require increasingly directional beamforming to establish and maintain the link. Intelligent beam management (BM) protocols will be critical for establishing and maintaining connections between the base station and user equipment in a dynamic channel. This article first provides a brief overview of the BM protocol in Release 15 of 5G New Radio, then identifies six major challenges to BM for later 5G releases that will likely persist into 6G. We describe the trends and issues behind each of the six challenges, and provide recommendations and suggested research directions to address them.