A unique experimental facility has been designed to measure damping of materials at cryogenic temperatures. The test facility pays special attention to removing other sources of damping in the measurement by avoiding frictional interfaces, decoupling the test specimen from the support system, and by using a non-contacting measurement device. Damping data is obtained for materials (Al, GrEp, Be, Fused Quartz), strain amplitudes (<10-6 ppm), frequencies (20 Hz-330Hz) and temperatures (20K - 293K) relevant to future precision optical space missions. The test data shows a significant decrease in viscous damping at cryogenic temperatures and can be as low as 10-4%, but the amount of the damping decrease is a function of frequency and material. Contrary to the other materials whose damping monotonically decreased with temperature, damping of Fused Quartz increased substantially at cryo, after reaching a minimum at around 150°K. The damping is also shown to be insensitive to strain for low strain levels. At room temperatures, the test data correlates well to the analytical predictions of the Zener damping model. Discrepancies at cryogenic temperatures between the model predictions and the test data are observed.