In this talk, we will discuss the structure preserving properties of the arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods. Based on the time dependent linear affiffiffine mapping, the ALE-DG methods presented here maintains almost all mathematical properties of DG methods on static grids, such as conservation, geometric conservation law, entropy stability and optimal error estimates. Meanwhile the mesh movement function requires only a very mild Lipschitz continuity and without any remapping. In this talk we will focus on the structure preserving property of the ALE-DG schemes for hyperbolic conservation laws, including the positivity preserving property for Euler equations and the well-balanced property for shallow water equations. The numerical stability, robustness and accuracy of the methods will also be shown by a variety of computational experiments on moving meshes.