Underwater images play a key role in ocean exploration but often suffer from severe quality degradation due to light absorption and scattering in water medium. Although major breakthroughs have been made recently in the general area of image enhancement and restoration, the applicability of new methods for improving the quality of underwater images has not specifically been captured. In this paper, we review the image enhancement and restoration methods that tackle typical underwater image impairments, including some extreme degradations and distortions. First, we introduce the key causes of quality reduction in underwater images, in terms of the underwater image formation model (IFM). Then, we review underwater restoration methods, considering both the IFM-free and the IFM-based approaches. Next, we present an experimental-based comparative evaluation of the state-of-the-art IFM-free and IFM-based methods, considering also the prior-based parameter estimation algorithms of the IFM-based methods, using both subjective and objective analyses (the used code is freely available at https://github.com/wangyanckxx/Single-Underwater-Image-Enhancement-and-Color-Restoration ). Starting from this paper, we pinpoint the key shortcomings of existing methods, drawing recommendations for future research in this area. Our review of underwater image enhancement and restoration provides researchers with the necessary background to appreciate challenges and opportunities in this important field.