This article proposes a novel regularization method, named spatio-spectral structure tensor total variation (S3TTV), for denoising and destriping hyperspectral (HS) images. HS images are inevitably contaminated by various types of noise during acquisition process due to the measurement equipment and the environment. For HS image denoising and destriping tasks, spatio-spectral total variation (SSTV) is widely known as a powerful regularization approach that models the spatio-spectral piecewise smoothness. However, since SSTV refers only to the local differences of pixels/bands, edges and textures that extend beyond adjacent pixels are not preserved during denoising process. To address this problem, we newly introduce S3TTV , which is designed to preserve two essential physical characteristics of HS images: semilocal spatial structures and spectral correlation across all bands. Specifically, we define S3TTV as the sum of the nuclear norms of spatio-spectral structure tensors, which are matrices formed by arranging second-order spatio-spectral difference vectors within semilocal areas. Furthermore, we formulate the HS image denoising and destriping problem as a constrained convex optimization problem involving S3TTV and develop an algorithm based on a preconditioned primal-dual splitting method to solve this problem efficiently. Finally, we demonstrate the effectiveness of S3TTV by comparing it with existing methods, including state-of-the-art ones through denoising and destriping experiments.