This chapter considers LQG optimal controls for input and state delayed systems. LQG controls use output feedback ones while LQ controls in the previous chapter require state feedback ones. State observers or filtered estimates are obtained from inputs and outputs to be used for LQG controls. First, finite horizon LQG controls are dealt with, which are fundamental results mathematically. Since they cannot be used as feedback controls due to the inherent requirement of infinite horizons associated with stability properties, infinite horizon LQG controls are obtained from these finite horizon LQG controls and discussed with their stability properties and some limitations. Then for general stabilizing feedback controls, receding horizon LQG controls, or model predictive LQG controls, are obtained from finite horizon LQG controls by the receding horizon concept, where their stability properties are discussed with some cost monotonicity properties. For input delayed systems, two different finite horizon LQG controls are obtained, one for a predictive LQG cost containing a state-predictor and the other for a standard LQG cost containing a state. They are obtained for free terminal states. State observers or filtered estimates are obtained from inputs and outputs for these two finite horizon LQG controls. From the finite horizon LQG controls, infinite horizon LQG controls are sought and discussed with stability properties and some limitations. Receding horizon LQG controls for input delayed systems are defined and obtained from the above two different finite horizon LQG controls. Appropriate state observers or filtered estimates such as standard Kalman filtered estimates and frozen-gain Kalman filtered estimates are discussed, which are used for LQG controls. Some conditions are investigated, under which the receding horizon LQG controls asymptotically stabilize the closed-loop systems. For state delayed systems, finite horizon LQG controls are obtained for two different LQG costs, one including single integral terminal terms and the other including double integral terminal terms. The solution is more complex as a cost becomes more complex. State observers or filtered estimates are obtained from inputs and outputs to be used for LQG controls. Then infinite horizon LQG controls are sought and discussed with their stability properties and some limitations. Receding horizon LQG controls for state delayed systems are obtained from the above finite horizon LQG controls for state delayed systems. Appropriate state observers or filtered estimates are discussed, which are used for LQG controls. Cost Monotonicity conditions are investigated, under which the receding horizon LQG controls asymptotically stabilize the closed-loop systems. It is shown that receding horizon LQG controls with the double integral terminal terms can have the delay-dependent stability condition while those with the single integral terminal terms have the delay-independent stability condition.