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Hyperspectral image (HSI) restoration is to recover the clean image from degraded version, such as the noisy, blurred, or damaged. Recent low-rank tensor-based recovery methods have been widely explored in HSIs restoration. Most of previous methods, however, neglect an inconspicuous but important phenomenon that the physical meaning and dimension along the spatial, spectral, and nonlocal mode are markedly different. In this work, we discover the low-rank property discrepancy along spatial, spectral, and nonlocal self-similarity mode in the HSIs, and argue that the intrinsic low-rank correlations along each mode contribute different to the final restoration results. Consequently, we figure out that the combination of the spectral and nonlocal-induced low-rank is most beneficial for HSIs modeling, and propose an optimal low-rank tensor (OLRT) model for HSIs restoration. Furthermore, we not only explore the low-rank property in the image component, but also in the sparse error component (stripe noise in HSIs). Thus, we extend OLRT to the OLRT-robust principal component analysis (RPCA) with low-rank tensor priors for both the HSIs and sparse error. Besides, previous methods are usually designed for one specific HSI task, which is less robust to various tasks. We prove that the proposed optimal low-rank prior is very flexible for various HSI restoration problems including denoising, deblurring, inpainting, and destriping. The proposed methods have been extensively evaluated on several benchmarks and tasks, and greatly outperform state-of-the-art (STOA). We show the simple yet effective OLRT strategy is also beneficial to STOA. [ABSTRACT FROM AUTHOR]