The CUR approximation of a matrix is appealing because, once the column and row indices are selected, it yields a low-rank approximation without even looking at the whole matrix. Remarkably, for any matrix there exists a CUR approximation that is near-best when compared with the truncated SVD. Efficient algorithms for computing CUR approximations have been developed, making them a competitive alternative in large-scale computations. In this talk, I will first discuss the computational and theoretical aspects of CUR, then highlight its power in practical applications, including approximation theory, model reduction and the solution of parameter-dependent problems.