The arbitrary Lagrangian–Eulerian (ALE) method has a wide range of applications in numerical simulation of multi-material fluid flow. The indirect ALE method consists of three steps: Lagrangian step, rezone step and remapping step. In this talk, we propose two classes of high order positivity-preserving conservative remapping methods on 2D and 3D meshes in the finite volume and discontinuous Galerkin (DG) frameworks respectively. Combined with the finite volume and DG Lagrangian schemes and the rezoning strategies, we present two types of high order positivity-preserving conservative ALE methods individually. For the finite volume framework, we adopt the multi-resolution WENO reconstruction which can achieve optimal accuracy in the smooth regions and keep non-oscillatory near discontinuities. Also we incorporate an efficient local limiting to preserve positivity for the positive physical variables involved in the ALE framework without sacrificing the original high-order accuracy and conservation. For the DG framework, we develop a high-order positivity-preserving polynomial projection remapping method based on the L2 projection for the DG scheme. A series of numerical tests are provided to verify properties of our remapping algorithms, such as high-order accuracy, conservation, essential non-oscillation, positivity-preserving and efficiency. The performance of the ALE methods using the above discussed remapping algorithms is also tested for the Euler system.