This paper considers the problem of designing a linear functional filter (LFF) to estimate a linear function of the state of a linear stochastic system. The LFF is a linear unbiased filter and is generally of lower order than the Kalman-Bucy filter. When the LFF is unbiased in the so-called linear-quadratic-Gaussian (LQG) problem, it is shown that the separation properties are established for the quadratic cost function and the stability. In the steady-state case, the minimal order, a canonical form and the optimal design are obtained for a single LFF which estimates a scalar-valued linear function of the state. Finally, an illustrative example is given to show the effectiveness of the LFF.