Adaptive grid method is one of the key techniques in the area of science and engineering applications. Based on the Lagrange grids of computational method, two types of different adaptive grid methods including couple adaptive mesh methods and pro-processor adaptive mesh methods are described. In the framework of couple adaptive mesh methods, several adaptive mesh methods including adaptive mesh refinement, moving mesh method, and adaptive coordinate methods are reviewed. In the framework of pro-processor adaptive mesh methods, adaptive mesh redistribution, reconnection-based method, changing connectivity of the mesh, and connectivity optimization method are described. Their advantages and disadvantages are compared with each other and analyzed with regard to the basic theory, efficiency, robustness for physics characteristics, and other features. It provides the support for using adaptive mesh method in the area of science and engineering applications.