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It is critical that the models pay attention not only to accuracy but also to the certainty of prediction. Uncertain predictions of deep models caused by noisy data raise significant concerns in trustworthy AI areas. To explore and handle uncertainty due to intrinsic data noise, we propose a novel method called ALUM to simultaneously handle the model uncertainty and data uncertainty in a unified scheme. Rather than solely modeling data uncertainty in the ultimate layer of a deep model based on randomly selected training data, we propose to explore mined adversarial triplets to facilitate data uncertainty modeling and non-parametric uncertainty estimations to compensate for the insufficiently trained latent model layers. Thus, the critical data uncertainty and model uncertainty caused by noisy data can be readily quantified for improving model robustness. Our proposed ALUM is model-agnostic which can be easily implemented into any existing deep model with little extra computation overhead. Extensive experiments on various noisy learning tasks validate the superior robustness and generalization ability of our method. The code is released at https://github.com/wwzjer/ALUM.