This paper considers the standard LQG problem on an infinite horizon with overtaking criterion. A control u*is overtaking optimal if for every other control u there is a time T‘u’ such that Cr‘u∗‘<Cr‘u’ for every T>T‘u’, where Cr‘u’ is the cost over the time interval [0, T] when employing u. We show that the separation principle holds and that the usual stationary optimal control is also optimal in the overtaking sense. We describe certain limiting properties of the optimal controlled process. The relation between overtaking optimal solutions and optimal solutions on finite large intervals is studied