In this paper we develop and analyze real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical testings demonstrate that new tilters significantly improve the extended Kalman filter with no additional cost and the new Gaussian sum filter has a nearly optimal performance.