Image restoration (IR) via deep learning has been vigorously studied in recent years. However, due to the ill-posed nature of the problem, it is challenging to recover the high-quality image details from a single distorted input especially when images are corrupted by multiple distortions. In this paper, we propose a multi-stage IR approach for progressive restoration of multi-degraded images via transferring similar edges/textures from the reference image. Our method, called a Reference-based Image Restoration Transformer (Ref-IRT), operates via three main stages. In the first stage, a cascaded U-Transformer network is employed to perform the preliminary recovery of the image. The proposed network consists of two U-Transformer architectures connected by feature fusion of the encoders and decoders, and the residual image is estimated by each U-Transformer in an easy-to-hard and coarse-to-fine fashion to gradually recover the high-quality image. The second and third stages perform texture transfer from a reference image to the preliminarily-recovered target image to further enhance the restoration performance. To this end, a quality-degradation-restoration method is proposed for more accurate content/texture matching between the reference and target images, and a texture transfer/reconstruction network is employed to map the transferred features to the high-quality image. Experimental results tested on three benchmark datasets demonstrate the effectiveness of our model as compared with other state-of-the-art multi-degraded IR methods. Our code and dataset are available at https://vinelab.jp/refmdir/.