To obtain high-visual-quality underwater images by image post-processing, many underwater image restoration and enhancement methods have been proposed. Underwater image quality assessment (UIQA) methods have been developed to compare these restoration and enhancement methods. This paper comprehensively summarizes the subjective and objective UIQA methods, metrics, and datasets. Experiments are conducted on two underwater image datasets to analyze the performance of several typical UIQA metrics. Suggestions for further research directions are put forward as well.