Three-Dimensional (3D) multiscale geometrical analysis is the technological fundamental for the processing of digital visual media, such as images, videos, and geometrical models. Its objective is to efficiently represent the point singularity, curve singularity, as well as surface singularity presented in those visual media. This study first reviews the research advances in two-dimensional (2D) multiscale geometrical analysis. It then elaborates on the development of 3D multiscale geometrical analysis for video according to the capability evolution in capturing singularity and nonlinear approximation efficiency improvement of various transforms. State-of-the-Art 3D multiscale geometrical analysis is classified into three categories: the extended multiscale geometrical analysis from 2D basis functions, the multiscale geometrical analysis based on 3D basis function, and the multiscale geometrical analysis based on spatiotemporal non-local correlation. The basic ideas of typical transforms are thoroughly discussed subsequently, and so are their nonlinear approximation efficiency, computational complexity, advantages, and disadvantages. Meanwhile, this study also presents a general review on the development of the 3D multiscale geometrical analysis for geometrical models. Based on the study above, the development trend of the 3D multiscale geometrical analysis is forecast in the near future.