The status of research on damping in fiber-reinforced composite materials and structures with emphasis on polymer composites has been reviewed in this paper. As a first step, composite damping mechanisms and methodology applicable to damping analysis is described. Further, the paper presents damping studies involving macromechanical, micromechanical and viscoelastic (relaxation and creep) approaches; models for interphase damping, damping and damage in composites. Some important works related to improved damping models for thick laminates, improvement of laminate damping and optimization for damping in fiber-reinforced composites/structures are critically reviewed.