The authors establish weighted L2-estimates of solutions for the damped wave equations with variable coefficients utt− divA(x)∇u+aut = 0 in IRn under the assumption a(x) ≥ a0[1+ρ(x)] −l, where a0 > 0, l < 1, ρ(x) is the distance function of the metric g = A −1(x) on IRn. The authors show that these weighted L2-estimates are closely related to the geometrical properties of the metric g = A −1(x).