Stochastic linear systems subject to time-varying parameter uncertainties affecting both system dynamics and noise statistics are considered. A linear filter is used to estimate a linear combination of the states of the system. When the filter is given, the stabilizing solution of a suitable Riccati equation is shown to yield an upper bound for the covariance of the estimator error. The main problem addressed in the paper is the design of an `optimal robust filter' that minimizes such a covariance bound. Necessary conditions are given for the existence of an optimal reduced-order robust filter as well as necessary and sufficient conditions for the full-order case. The computation of the optimal filter calls for the solution of a Riccati equation that generalizes the standard Riccati equation for the Kalman filtering problem. A numerical example is provided in which the new filter is compared with both the Kalman and H-infinity-filters.