Abstract: This paper provides a general framework solution to state estimation of Gaussian filter for both linear and nonlinear dynamic systems with multiple step randomly delayed measurements. Noise and previous state vectors are added into the current state vector to facilitate its recursive update estimation. A general framework of Bayesian solution to the augmented state estimation is then derived. For nonlinear systems, different Gaussian approximation filters can be developed by utilizing different numerical methods for computing Gaussian weighted integrals involved in the Bayesian solution. Finally, the third-degree spherical-radial cubature rule is used to implement the proposed method. Simulation is performed based on a target tracking model, in which measurements are randomly delayed for multiple steps. The simulation results illustrate the efficiency and advantages of the proposed method.