The most widely used filter to estimate the state of a nonlinear stochastic system from noisy observation data is the extended Kalman filter. However, if the nonlinearities are significant, its performance can be considerably improved as recent works by Alspace and Sorenson (1967, 1972), C. P. Fang, Julier and Uhlmann (1994, 1995) have shown. The authors of the present paper continue these efforts by developing and analyzing real-time and accurate filters for nonlinear filtering algorithms based on Gaussian distributions. Their paper presents a systematic formulation of Gaussian filters and mixed Gaussian filters. The proposed Gaussian filter is based on two steps: the conditional probability density is supposed to exist and to be a Gaussian distribution; the filter is obtained by equating the Bayesian formula w.r.t. the first two moments. The approach is based on the efficient numerical integration of the Bayesian formula for optimal recursive filtering. Furthermore, the authors also discuss mixed Gaussian filters in which the conditional probability density is approximated by sums of Gaussian distributions. The Gaussian sum filter, already studied by Alspace and Sorenson in 1967, 1972, is adapted for the update of Gaussian distributions and new update rules of weights of Gaussian sum filters are proposed. Through simulations the authors show that the filters developed in their paper have superior performance to the filter of Julier-Uhlmann (1994) and the extended Kalman filter.