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 proposed filter. We also discuss mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. Our numerical testings demonstrate that new filters significantly improve the extended Kalman filter with no additional cost and the new Gaussian sum filter has a nearly optimal performance.