Highly effective vibration damping materials for metal sheet structures can be obtained by making use of amorphous high polymers in the viscoelastic state. Materials of this kind can be adjusted to provide optimum damping at given frequencies in temperature bands of prescribed position and width. This can be done by mixing high polymers having different glass transition temperatures in suitable distribution or by copolymerizing suitable monomers. Plastification and addition of filling materia[s can play a part in obtaining the desired results. The larger the temperature bandwidth of high internal energy losses the larger is the frequency bandwidth. However, the maximum damping will be diminished when the bandwidth is increased; this follows from the laws which govern the molecular behavior of high polymers. Damping materials have been developed for different combined systems, such as metal sheets with a single damping layer at one side or sandwich systems, according to special technical demands with respect to frequency and temperature dependence. Experimental results obtained with these combined systems will illustrate the damping efficiencies which can be attained in optimum cases.