Abstract: Diffusion-based image super-resolution (SR) models have attracted substantial interest due to their powerful image restoration capabilities. However, prevailing diffusion models often struggle to strike an optimal balance between efficiency and performance. Typically, they either neglect to exploit the potential of existing extensive pretrained models, limiting their generative capacity, or they necessitate a dozens of forward passes starting from random noises, compromising inference efficiency. In this paper, we present DoSSR, a $\textbf{Do}$main $\textbf{S}$hift diffusion-based SR model that capitalizes on the generative powers of pretrained diffusion models while significantly enhancing efficiency by initiating the diffusion process with low-resolution (LR) images. At the core of our approach is a domain shift equation that integrates seamlessly with existing diffusion models. This integration not only improves the use of diffusion prior but also boosts inference efficiency. Moreover, we advance our method by transitioning the discrete shift process to a continuous formulation, termed as DoS-SDEs. This advancement leads to the fast and customized solvers that further enhance sampling efficiency. Empirical results demonstrate that our proposed method achieves state-of-the-art performance on synthetic and real-world datasets, while notably requiring $\textbf{\emph{only 5 sampling steps}}$. Compared to previous diffusion prior based methods, our approach achieves a remarkable speedup of 5-7 times, demonstrating its superior efficiency.