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Remote sensing image super-resolution (RSISR) techniques play a crucial role in various remote sensing applications. However, deep learning-based methods applied to RSISR encounter difficulties in learning complex features of remote sensing images and modeling long-term correlations between pixels. This study proposes a multiscale sliding window attention generation adversarial network (MSWAGAN), which combines the advantages of convolutional neural networks (CNNs) and transformers to overcome these limitations. The MSWAGAN consists of three main parts. In the shallow feature extraction part, CNN is used to extract shallow features from remote sensing images. The deep feature extraction part is divided into two stages. First, a multiscale sliding window attention (MSWA) is designed to replace the multihead attention (MHA) in the transformer. MSWA can learn local multiscale complex features of remote sensing images without increasing the number of parameters in MHA. Then, the transformer is utilized to learn global image features and model the long-range correlations between pixels. The image reconstruction part utilizes subpixel convolution for feature upsampling. Furthermore, in order to extend the application of super-resolution (SR) remote sensing images, a cross-sensor real multispectral RSISR dataset consisting of Landsat-8 (L8) and Sentinel-2 (S2) images was constructed, and a series of experiments to improve the spatial resolution of L8 images from 30 to 10 m in B, G, R, and near-infrared (NIR) bands were conducted. Experimental results demonstrate that our method outperforms some of the latest SR methods.