Multiview subspace learning has attracted much attention due to the efficacy of exploring the information on multiview features. Most existing methods perform data reconstruction on the original feature space and thus are vulnerable to noisy data. In this article, we propose a novel multiview subspace learning method, called multiview consensus structure discovery (MvCSD). Specifically, we learn the low-dimensional subspaces corresponding to different views and simultaneously pursue the structure consensus over subspace clustering for multiple views. In such a way, latent subspaces from different views regularize each other toward a common consensus that reveals the underlying cluster structure. Compared to existing methods, MvCSD leverages the consensus structure derived from the subspaces of diverse views to better exploit the intrinsic complementary information that well reflects the essence of data. Accordingly, the proposed MvCSD is capable of producing a more robust and accurate representation structure which is crucial for multiview subspace learning. The proposed method can be optimized effectively, with theoretical convergence guarantee, by alternatively iterating the argument Lagrangian multiplier algorithm and the eigendecomposition. Extensive experiments on diverse datasets demonstrate the advantages of our method over the state-of-the-art methods.