Multiview spectral clustering (MVSC) has achieved state-of-the-art clustering performance on multiview data. Most existing approaches first simply concatenate multiview features or combine multiple view-specific graphs to construct a unified fusion graph and then perform spectral embedding and cluster label discretization with k -means to obtain the final clustering results. They suffer from an important drawback: all views are treated as fixed when fusing multiple graphs and equal when handling the out-of-sample extension. They cannot adaptively differentiate the discriminative capabilities of multiview features. To alleviate these problems, we propose a flexible MVSC with self-adaptation (FMSCS) method in this article. A self-adaptive learning scheme is designed for structured graph construction, multiview graph fusion, and out-of-sample extension. Specifically, we learn a fusion graph with a desirable clustering structure by adaptively exploiting the complementarity of different view features under the guidance of a proper rank constraint. Meanwhile, we flexibly learn multiple projection matrices to handle the out-of-sample extension by adaptively adjusting the view combination weights according to the specific contents of unseen data. Finally, we derive an alternate optimization strategy that guarantees desirable convergence to iteratively solve the formulated unified learning model. Extensive experiments demonstrate the superiority of our proposed method compared with state-of-the-art MVSC approaches. For the purpose of reproducibility, we provide the code and testing datasets at https://github.com/shidan0122/FMICS.