Rank aggregation, which combines multiple individual rank lists toobtain a better one, is a fundamental technique in various applications such as meta-search and recommendation systems. Most existing rank aggregation methods blindly combine multiple rank lists with possibly considerable noises, which often degrades their performances. In this paper, we propose a new model for robust rank aggregation (RRA) via matrix learning, which recovers a latent rank list from the possibly incomplete and noisy input rank lists. In our model, we construct a pairwise comparison matrix to encode the order information in each input rank list. Based on our observations, each comparison matrix can be naturally decomposed into a shared low-rank matrix, combined with a deviation error matrix which is the sum of a column-sparse matrix and a row-sparse one. The latent rank list can be easily extracted from the learned low-rank matrix. The optimization formulation of RRA has an element-wise multiplication operator to handle missing values, a symmetric constraint on the noise structure, and a factorization trick to restrict the maximum rank of the low-rank matrix. To solve this challenging optimization problem, we propose a novel procedure based on the Augmented Lagrangian Multiplier scheme. We conduct extensive experiments on meta-search and collaborative filtering benchmark datasets. The results show that the proposed RRA has superior performance gain over several state-of-the-art algorithms for rank aggregation.