Various greedy algorithms have been developed for sparse signal recovery in recent years. However, most of them utilize the \ell-{2} norm based loss function and sensitive to non-Gaussian noises and outliers. This paper proposes a Cauchy matching pursuit (CauchyMP) algorithm for robust sparse representation and classification. By leveraging a Cauchy estimator based loss function, the proposed approach can robustly learn the sparse representation of noisy data corrupted by various severe noises. As a greedy algorithm, CauchyMP is also computationally efficient. We also develop a CauchyMP based classifier for robust classification with application to face recognition. The experiments on the datasets with gross corruptions demonstrate the efficacy and robustness of CauchyMP for learning robust sparse representation.