A tuned mass damper (TMD) is a device consisting of a mass, a spring, and a damper attached to a structure to reduce its dynamic response. The TMD concept was first applied at the beginning of the 20th century to reduce the rolling motion of ships and ship hull vibrations. Although currently they are mainly used to reduce wind-induced oscillations in buildings, there are also applications to reduce the response of the building under seismic action. Plastic deformations induced in the structural elements during strong earthquakes modify the dynamic characteristics of the structure, which makes the behaviour of the TMD system during seismic action more complex than in the case of wind, where the structure remains elastic. Often, the performance requirement that dictates the design of a building structure is the limitation of drifts induced by frequent earthquakes or by wind. Although the TMD may not be practical in reducing the structural response under larger earthquakes, it may represent a convenient solution for limiting non-structural damage under frequent earthquakes. To investigate this possibility, a steel moment resisting frame equipped with a sliding TMD system is designed and analysed using nonlinear response history analyses. The advantages, disadvantages, and effectiveness of TMD systems in reducing structural response in frequent and design-level earthquakes are investigated.