Damping is an important aspect of aerospace structural dynamics. Passive damping is caused by numerous physical mechanisms, only some of which can be exploited by design. The most effective methods of designing materials‐based passive damping into structures involve the use of: high damping viscoelastic polymers in layered damping treatments; elastomers; shunted piezoelectrics; and damped composites. To consider damping during the design process, accurate models are needed—these must capture the general effects of damping, while also enabling the detailed design of damping treatments. Strain‐based viscous damping models yield modal damping that increases monotonically with frequency, a result inconsistent with observations. More general proportional damping is best regarded as a mathematical curiosity. Complex stiffness models, although capable of accommodating frequency‐dependent damping and stiffness in frequency response analysis, cannot directly predict structural response to arbitrary loading. The combination of either the complex stiffness or the modal strain energy method with a modal damping model, however, often yields acceptable results. The need for better accuracy in some applications has motivated the development of improved time‐domain models that capture the frequency‐, temperature‐, and amplitude dependence of damping and stiffness exhibited by the high‐loss materials that are frequently used to effect significant damping.