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Extensive attention has been widely paid to enhance the spatial resolution of hyperspectral (HS) images with the aid of multispectral (MS) images in remote sensing. However, the ability in the fusion of HS and MS images remains to be improved, particularly in large-scale scenes, due to the limited acquisition of HS images. Alternatively, we super-resolve MS images in the spectral domain by the means of partially overlapped HS images, yielding a novel and promising topic: spectral superresolution (SSR) of MS imagery. This is challenging and less investigated task due to its high ill-posedness in inverse imaging. To this end, we develop a simple but effective method, called joint sparse and low-rank learning (J-SLoL), to spectrally enhance MS images by jointly learning low-rank HS-MS dictionary pairs from overlapped regions. J-SLoL infers and recovers the unknown hyperspectral signals over a larger coverage by sparse coding on the learned dictionary pair. Furthermore, we validate the SSR performance on three HS-MS datasets (two for classification and one for unmixing) in terms of reconstruction, classification, and unmixing by comparing with several existing state-of-the-art baselines, showing the effectiveness and superiority of the proposed J-SLoL algorithm. Furthermore, the codes and datasets will be available at: https://github.com/danfenghong/IEEE\_TGRS\_J-SLoL, contributing to the RS community.