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• We proposed a novel Robust Matrix Completion method based on Rank-One matrix approximation (RMCRO), which divides the noisy and incomplete data matrix into two parts: a low-rank matrix and a sparse matrix, and it is robust to sparse noise. • We use a rank-one matrix pursuit method to approximate the low-rank part of the incomplete data matrix. Therefore, the proposed algorithm can recover the missing entries of the incomplete data matrix and estimate the rank simultaneously. • Extensive experimental results on both synthetic datasets and real world datasets demonstrate the effectiveness of the proposed method for matrix completion and rank estimation. Matrix completion aims at estimating the missing entries of a low-rank and incomplete data matrix. It frequently arises in many applications such as computer vision, pattern recognition, recommendation system, and data mining. Most of the existing methods face two problems. Firstly, the data matrix in real world is often disturbed by noise. Noise may change the date structure of the incomplete matrix, thereby degrade the performance of matrix completion algorithms. Secondly, some existing methods need to preset a reasonable rank as input, and the value of rank will affect the performance of the algorithms. Therefore, we proposed a robust rank-one matrix completion method with rank estimation in this paper. To mitigate the influence of noise, we divide the incomplete and noisy data matrix into two parts iteratively: low-rank and sparse parts. Besides, we use a weighted rank-one matrix pursuit algorithm to approximate the low-rank part of the data matrix, and the rank of the matrix can be estimated with the adaptive weight vector. The performance of the proposed method is demonstrated by experiments on both synthetic datasets and image datasets. The experimental results demonstrate the performance of the proposed method with incompleted matrices distrubed by sparse noise. [ABSTRACT FROM AUTHOR]