Autonomous navigation and positioning are key to the successful performance of unmanned underwater vehicles (UUVs) in environmental monitoring, oceanographic mapping, and critical marine infrastructure inspections in the sea. Cameras have been at the center of attention as an underwater sensor due to the advantages of low costs and rich content information in high visibility ocean waters, especially in the fields of underwater target recognition, navigation, and positioning. This paper is not only a literature overview of the vision-based navigation and positioning of autonomous UUVs but also critically evaluates the methodologies which have been developed and that directly affect such UUVs. In this paper, the visual navigation and positioning algorithms are divided into two categories: geometry-based methods and deep learning-based. In this paper, the two types of SOTA methods are compared experimentally and quantitatively using a public underwater dataset and their potentials and shortcomings are analyzed, providing a panoramic theoretical reference and technical scheme comparison for UUV visual navigation and positioning research in the highly dynamic and three-dimensional ocean environments.