<p>Non-Gaussian noise and measurement anomalies are major challenges in using filters for state estimation in engineering practice, particularly in the absence of relevant prior statistical information. Hence, we propose an adaptive-weighted variational Bayesian filter (AW-VBF) to obtain a robust state estimation. First, the variational Bayes approach is used to establish a fundamental filter framework that can handle both nonlinear and non-Gaussian scenarios by approximating the true posterior distribution through a parameterized distribution. Interestingly, the state transition and state measurement processes play independent roles in this framework. Subsequently, kernel density estimation is adopted to capture non-Gaussian noise characteristics from historical data. A novel adaptive weight function that is twice differentiable (thus ensuring the existence of gradient estimators) replaces the likelihood loss function to address measurement anomalies. The algorithm process, including the gradient estimator details, is provided. Target-tracking simulation results under different conditions verify the superiority of the AW-VBF to existing methods. Compared with these conventional methods, our method enhances position estimation accuracy by 37.96% and velocity estimation accuracy by 32.35% in the presence of non-Gaussian noise. The corresponding enhancements in the presence of measurement anomalies are 64.92% and 25.59%, respectively.</p>