This paper addresses the design of hybrid digital and analog beamforming for large-scale antenna arrays in millimeter wave (mmWave) communication systems. The goal is to achieve performance close to fully digital beamforming with significantly fewer radio frequency (RF) chains. The hybrid architecture consists of a low-dimensional digital beamformer followed by an RF beamformer using analog phase shifters. The paper establishes that with twice the number of RF chains as data streams, the hybrid structure can exactly replicate any fully digital beamformer. For cases with fewer RF chains, the paper proposes heuristic algorithms for both point-to-point MIMO and downlink multi-user MISO systems. These algorithms aim to maximize spectral efficiency under power constraints. The paper also considers practical scenarios with finite resolution phase shifters and shows that the proposed schemes perform well even with low-resolution phase shifters. The algorithms are modified for finite resolution phase shifters, and numerical simulations demonstrate their effectiveness. The paper also discusses the design of hybrid beamformers for multi-user massive MISO systems, where the RF precoder is iteratively designed to optimize the weighted sum rate. The results show that adding more RF chains can reduce the gap between achievable rates and capacity. The proposed algorithms are efficient and effective for large-scale antenna arrays in mmWave systems.This paper addresses the design of hybrid digital and analog beamforming for large-scale antenna arrays in millimeter wave (mmWave) communication systems. The goal is to achieve performance close to fully digital beamforming with significantly fewer radio frequency (RF) chains. The hybrid architecture consists of a low-dimensional digital beamformer followed by an RF beamformer using analog phase shifters. The paper establishes that with twice the number of RF chains as data streams, the hybrid structure can exactly replicate any fully digital beamformer. For cases with fewer RF chains, the paper proposes heuristic algorithms for both point-to-point MIMO and downlink multi-user MISO systems. These algorithms aim to maximize spectral efficiency under power constraints. The paper also considers practical scenarios with finite resolution phase shifters and shows that the proposed schemes perform well even with low-resolution phase shifters. The algorithms are modified for finite resolution phase shifters, and numerical simulations demonstrate their effectiveness. The paper also discusses the design of hybrid beamformers for multi-user massive MISO systems, where the RF precoder is iteratively designed to optimize the weighted sum rate. The results show that adding more RF chains can reduce the gap between achievable rates and capacity. The proposed algorithms are efficient and effective for large-scale antenna arrays in mmWave systems.