Low-rank-tensor-approximation (LRTA)-based hyperspectral image and hyperspectral imagery (HSI) restoration has drawn increasing attention. However, most of the methods construct a hidden low-rank tensor by utilizing the nonlocal self-similarity (NLSS) and global spectral correlation (GSC) inherited by HSIs. Although achieving state-of-the-art (SOTA) restoration performance, NLSS and GSC have limitations. NLSS is introduced from natural image denoising to remove spatially independent identically distributed (i.i.d.) Gaussian and impulse noise, while GSC, which is naturally possessed by HSIs, is adopted to maintain the spectral integrity and remove spectrally, i.i.d., degradations. Therefore, NLSS and GSC may not be successfully used for complex HSI restoration tasks, such as destriping, cloud removal, and recovery of atmospheric absorption bands. To solve the issue, borrowing the idea from manifold learning, the geometry information characterized by proximity relationship, is integrated with the LRTA to solve the above issue, named multigraph-based LRTA (MGLRTA). Different from most of the existing methods, the proposed MGLRTA directly models an HSI as a low-rank tensor and efficiently explores the extra proximity information on the defined graphs that are not only inherited by the low-rank constraints but also naturally possessed in HSIs. A well-posed iterative algorithm is designed to solve the restoration problem. Experimental results on different datasets that cover several severe degradation scenarios demonstrate that the proposed MGLRTA outperforms the SOTA HSI restoration methods.