Nonnegative matrix factorization (NMF) related models are now widely used in text mining, bioinformatics, knowledge transfer, recommender systems, semi-supervised and unsupervised learning. However so far, the basic models use least square formulation which is prone to large noise and outliers. After reviewing major NMF models, here we present robust NMF models using L1 and L21 norms which exhibit stability and robustness w.r.t. large noises. We present computational algorithms for these models with rigorous theoretical analysis. These algorithm are as efficient as the algorithms for least square formulations, avoiding the significant computational complexities routinely associated with L1, L21 formulations. Experiments in image data demonstrate the strong robustness of the robust NMF models.