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Image restoration and enhancement is a process of improving the image quality by removing degradations, such as noise, blur, and resolution degradation. Deep learning (DL) has recently been applied to image restoration and enhancement. Due to its ill-posed property, plenty of works have been explored priors to facilitate training deep neural networks (DNNs). However, the importance of priors has not been systematically studied and analyzed by far in the research community. Therefore, this paper serves as the first study that provides a comprehensive overview of recent advancements in priors for deep image restoration and enhancement. Our work covers five primary contents: (1) A theoretical analysis of priors for deep image restoration and enhancement; (2) A hierarchical and structural taxonomy of priors commonly used in the DL-based methods; (3) An insightful discussion on each prior regarding its principle, potential, and applications; (4) A summary of crucial problems by highlighting the potential future directions, especially adopting the large-scale foundation models as prior, to spark more research in the community; (5) An open-source repository that provides a taxonomy of all mentioned works and code links.