Aiming at the online learning with outliers, this paper proposes a robust regularized online sequential extreme learning machine (RR-OSELM). The proposed RR-OSELM is able to learn the newly arrived samples incrementally by a recursive fashion, and assign inverse weights for each example based on the priori error so as to reduce its sensibility to outliers. The Tikhonov regularization technique is incorporated in the RR-OSELM to further enhance the stability of the algorithm in real applications. Experimental results show that the proposed RR-OSELM is more robust than its counterparts, and it can be applied to the online modeling and prediction of data streams with outliers.