Vibrating structures, such as gas turbine engines, are assemblages of multiple components interfaced with each other through physical contact. The friction produced at these contact interfaces, by sliding or slipping at the interface, is a source of energy dissipation, resulting in damping effects. This passive approach to damping is exploited by engineers to provide optimal energy dissipation to attenuate vibration in a variety of systems (such as bladed disc assemblies and airfoils), to prevent wear and premature failure. However, the highly nonlinear nature of friction renders accurate predictive modeling of such a problem challenging. This article gives an overview of the current state of the art, and the methods used for mathematical and predictive computational modeling of vibrating systems under the influence of dry friction.