This talk concerns the asymptotic stability for the damped oscillators with multiple degrees of freedom, x′′ + h(t)x′ + Ax = 0 and its generalization Mx′′ + C(t)x′ + Kx = 0; where h(t) is a nonnegative function for t 0, A, M and K are n n real constant matrices. and C(t) is an n n matrix whose elements are real-valued functions of t. The functions h(t) and C(t) correspond to the damping coefficient and the damping matrix, respectively. The origin (x; x′) = (0; 0) is the only equilibrium of the above-mentioned damped oscillators. Necessary and sufficient conditions are presented for the equilibrium of these oscillators to be asymptotically stable. The obtained conditions are given by the forms of certain growth conditions concerning the damping h(t) and C(t), respectively.