The authors consider the problem of controlling a discrete-time linear system by output feedback so as to have a second output z(t) track an observed reference signal tau(t). First, as a preliminary, we consider the problem of asymptotic tracking, i.e., to design a regulator such that \z(t) - tau(t)\ --> 0. This problem has been studied intensely in the literature, mainly in the continuous-time case. It is known that only under very special conditions does there exist a linear regulator which achieves this design goal and which is universal in the sense that it works for all reference signals and does not depend on them. On the other hand, if tau(t) is a harmonic signal with known frequencies but with unknown amplitudes and phases, there exist such regulators under mild conditions, provided the dimension of tau(t) is no larger than the number of controls. This is true even if the plant itself is corrupted by an unobserved additive harmonic disturbance wt of the same type as tau(t), if the dimension of w(t) is no larger than the number of outputs available for feedback control. However, if the first dimensionality condition is not satisfied, asymptotic tracking is not possible, but a steady-state tracking error remains. Therefore, the authors turn to another approach to the tracking problem, which also allows for damping of other system and control variables, and this is our main result. The measure of performance is given by a natural quadratic cost function. The object is to design an optimal regulator which is universal in the sense that it does not depend on the unknown amplitudes and phases of tau(t) and w(t) and is optimal for all choices of tau(t) and w(t). The authors prove that an optimal universal regulator exists in a wide class of stabilizing and possibly nonlinear regulators under natural technical conditions and that this regulator is in fact linear, provided that the second dimensionality condition above is satisfied. On the other hand, if it is not satisfied, the existence of an optimal universal regulator is not a generic property, so as a rule no optimal universal regulator exists. The authors provide complete solutions of all the problems described above.