Abstract: Digital geometry processing focus on 2-dimensional surfaces in 3-dimentional space.Laplace-Beltrami operator is a differential operator defined on Riemannian manifolds.Discrete Laplace-Beltrami operator is a useful tool in applications such as 3-dimentional model analysis.Diverse methods of discretization lead to distinct mathematical properties related to the applications they aim at respectively.This paper provides a survey on theory and application of discrete Laplace-Beltrami operator.We hope this paper can provide some help for researchers to learn how Laplace-Beltrami operator works in digital geometry processing with in-depth understanding and identify possible directions for further research and new applications.