This paper is motivated by the filtering estimation for a class of nonlinear stochastic systems in the case that the measurements are randomly delayed by one sampling time. Through presenting Gaussian approximation about the one-step posterior predictive probability density functions (PDFs) of the state and delayed measurement, a novel Gaussian approximation (GA) filter is derived, which recursively operates by analytical computation and Gaussian weighted integrals. The proposed GA filter gives a general and common framework since: (1) it is applicable for both linear and nonlinear systems, (2) by setting the delay probability as zero, it automatically reduces to the standard Gaussian filter without the randomly delayed measurements, and (3) many variations of the proposed GA filter can be developed through utilizing different numerical technologies for computing such Gaussian weighted integrals, including the previously existing EKF and UKF methods, as well as the improved cubature Kalman filter (CKF) in our paper using the spherical-radial cubature rule. The performance of the new method is demonstrated with a simulation example of the high-dimensional GPS/INS integrated navigation. (C) 2013 Elsevier Ltd. All rights reserved.