<p>Measurements and dynamic process models are often contaminated by errors or noises. In many real applications, for simplicity of computations, it is assumed such errors are random and follow a normal distribution. However, the normal-distribution assumption can only be an approximation; and the true distribution can be a heavier tailed one rather than the assumed normal one. Some optimal estimators, including Kalman filters, can face serious performance degradation in the presence of heavy-tailed distributions. Previous robust Kalman filtering schemes are not directly targeted on these two types of noises, especially the process noise; instead, the predicted state estimation error is considered together with the measurement noise. The topic of the current study is the robust modification of the Kalman filtering against heavy-tailed distributions of both the process and measurement noises. A new robust Kalman filter is proposed. The new filter is simple, in the sense that it has the same structure of a standard Kalman filter. It is highly automated, in the sense that it can be initialized by the standard Kalman filtering. It is effective as it is insensitive to heavy-tailed distributions of both the process and measurement noises. The proposed method is general and flexible, in the sense that it can be readily modified to consider reliable prior distributional information. Simulation of a tracking application and real data analysis of an integrated GNSS and INS application demonstrate the effectiveness of the proposed method. The tracking simulation shows the proposed method reduces the positioning Root Mean Squared Error (RMSE) by 11.1% compared with the standard Kalman filter, and the reduction can reach 20.0% with five iterations. The real GNSS/INS navigation experiment achieves up to 22% reduction in positioning RMSE and obvious improvements in velocimetry performance relative to the standard Kalman filter.</p>