This paper presents low-complexity channel estimation and hybrid analog/digital precoding algorithms for millimeter wave (mmWave) cellular systems. The algorithms exploit the sparse nature of mmWave channels and the limited scattering in these systems. The proposed channel estimation algorithm uses an adaptive compressed sensing (CS) approach with a hierarchical multi-resolution codebook to efficiently estimate mmWave channel parameters. The codebook is designed to generate beamforming vectors with different beamwidths, enabling the algorithm to adaptively estimate channel parameters with high success probability. The algorithm is shown to achieve comparable precoding gains to exhaustive channel training methods, and to approach the performance of digital solutions even in the presence of interference. The hybrid precoding algorithm proposed in the paper overcomes hardware constraints on analog-only beamforming and achieves performance close to digital solutions. Simulation results demonstrate that the proposed algorithms achieve high spectral efficiency and coverage probability, comparable to systems with perfect channel knowledge. The paper also provides a detailed system model and formulation of the mmWave channel estimation problem, and presents the design of a hybrid analog/digital codebook for beamforming vectors. The algorithms are shown to be effective in both single-path and multi-path mmWave channels, and the paper provides insights into the efficient allocation of training power among the adaptive stages of the algorithm.This paper presents low-complexity channel estimation and hybrid analog/digital precoding algorithms for millimeter wave (mmWave) cellular systems. The algorithms exploit the sparse nature of mmWave channels and the limited scattering in these systems. The proposed channel estimation algorithm uses an adaptive compressed sensing (CS) approach with a hierarchical multi-resolution codebook to efficiently estimate mmWave channel parameters. The codebook is designed to generate beamforming vectors with different beamwidths, enabling the algorithm to adaptively estimate channel parameters with high success probability. The algorithm is shown to achieve comparable precoding gains to exhaustive channel training methods, and to approach the performance of digital solutions even in the presence of interference. The hybrid precoding algorithm proposed in the paper overcomes hardware constraints on analog-only beamforming and achieves performance close to digital solutions. Simulation results demonstrate that the proposed algorithms achieve high spectral efficiency and coverage probability, comparable to systems with perfect channel knowledge. The paper also provides a detailed system model and formulation of the mmWave channel estimation problem, and presents the design of a hybrid analog/digital codebook for beamforming vectors. The algorithms are shown to be effective in both single-path and multi-path mmWave channels, and the paper provides insights into the efficient allocation of training power among the adaptive stages of the algorithm.