In this note, we shall present a new class of Gaussian filters called Fourier-Hermite Kalman filters. Fourier-Hermite Kalman filters are based on expansion of nonlinear functions with the Fourier-Hermite series in same way as the traditional extended Kalman filter is based on the Taylor series. The first order truncation of the Fourier-Hermite series gives the previously known statistically linearized filter.