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Magnetic resonance imaging (MRI) can present multi-contrast images of the same anatomical structures, enabling multi-contrast super-resolution (SR) techniques. Compared with SR reconstruction using a single-contrast, multi-contrast SR reconstruction is promising to yield SR images with higher quality by leveraging diverse yet complementary information embedded in different imaging modalities. However, existing methods still have two shortcomings: (1) they neglect that the multi-contrast features at different scales contain different anatomical details and hence lack effective mechanisms to match and fuse these features for better reconstruction; and (2) they are still deficient in capturing long-range dependencies, which are essential for the regions with complicated anatomical structures. We propose a novel network to comprehensively address these problems by developing a set of innovative Transformer-empowered multi-scale contextual matching and aggregation techniques; we call it McMRSR. Firstly, we tame transformers to model long-range dependencies in both reference and target images. Then, a new multi-scale contextual matching method is proposed to capture corresponding contexts from reference features at different scales. Furthermore, we introduce a multi-scale aggregation mechanism to gradually and interactively aggregate multi-scale matched features for reconstructing the target SR MR image. Extensive experiments demonstrate that our network outperforms state-of-the-art approaches and has great potential to be applied in clinical practice. Codes are available at https://github.com/XAIMI-Lab/McMRSR.