Samples collected from most industrial processes have two challenges: one is contaminated by the non-Gaussian noise, and the other is gradually obsolesced. This feature can obviously reduce the accuracy and generalization of models. To handle these challenges, a novel method, named the robust online extreme learning machine (RO-ELM), is proposed in this paper, in which the least mean
\boldsymbolp-power criterion is employed as the cost function which is to boost the robustness of the ELM, and the forgetting mechanism is introduced to discard the obsolescence samples. To investigate the performance of the RO-ELM, experiments on artificial and real-world datasets with the non-Gaussian noise are performed, and the datasets are from regression or classification problems. Results show that the RO-ELM is more robust than the ELM, the online sequential ELM (OS-ELM) and the OS-ELM with forgetting mechanism (FOS-ELM). The accuracy and generalization of the RO-ELM models are better than those of other models for online learning.