Delay alignment modulation (DAM) is a novel wideband transmission technique for mmWave massive MIMO systems, which exploits the high spatial resolution and multi-path sparsity to mitigate ISI, without relying on channel equalization or multi-carrier transmission. In particular, DAM leverages the delay pre-compensation and path-based beamforming to effectively align the multi-path components, thus achieving the constructive multi-path combination for eliminating the ISI while preserving the multi-path power gain. Different from the existing works only considering single-user DAM, this paper investigates the DAM technique for multi-user mmWave massive MIMO communication. First, we consider the asymptotic regime when the number of antennas Mt at BS is sufficiently large. It is shown that by employing the simple delay pre-compensation and per-path-based MRT beamforming, the single-carrier DAM is able to perfectly eliminate both ISI and IUI. Next, we consider the general scenario with Mt being finite. In this scenario, we characterize the achievable rate region of the multi-user DAM system by finding its Pareto boundary. Specifically, we formulate a rate-profile-constrained sum rate maximization problem by optimizing the per-path-based beamforming. Furthermore, we present three low-complexity per-path-based beamforming strategies based on the MRT, zero-forcing, and regularized zero-forcing principles, respectively, based on which the achievable sum rates are studied. Finally, we provide simulation results to demonstrate the performance of our proposed strategies as compared to two benchmark schemes based on the strongest-path-based beamforming and the prevalent OFDM, respectively. It is shown that DAM achieves higher spectral efficiency and/or lower peak-to-average-ratio, for systems with high spatial resolution and multi-path diversity.