We investigate beam training and allocation for multiuser millimeter wave massive MIMO systems. An orthogonal pilot based beam training scheme is first developed to reduce the number of training times, where all users can simultaneously perform the beam training with the base station (BS). As the number of users increases, the same beam from the BS may point to different users, leading to beam conflict and multiuser interference. Therefore, a quality-of-service (QoS) constrained (QC) beam allocation scheme is proposed to maximize the equivalent channel gain of the QoS-satisfied users, under the premise that the number of the QoS-satisfied users without beam conflict is maximized. To reduce the overhead of beam training, two partial beam training schemes, an interlaced scanning (IS) and a selection probability (SP) based schemes, are proposed. The overhead of beam training for the IS-based scheme can be reduced by nearly half while the overhead for the SP-based scheme is flexible. Simulation results show that the QC-based beam allocation scheme can effectively mitigate the interference caused by the beam conflict and significantly improve the spectral efficiency while the IS-based and SP-based schemes significantly reduce the overhead of beam training at the cost of sacrificing spectral efficiency a little.