In this talk, the high order multi-resolution weighted essentially non-oscillatory (WENO) schemes for Hamilton-Jacobi (HJ) equations on moving meshes are discussed. First, we will develop a high order arbitrary Lagrangian-Eulerian (ALE) WENO finite difference method for HJ equations, which is based on moving quadrilateral meshes. Next, we are concerned with the study of efficient and high order accurate numerical methods for solving HJ equations with initial conditions defined in the whole domain. Specifically, we use the above mentioned high order ALE-WENO finite difference scheme in a finite and moving computational domain, with numerical boundary conditions obtained by solving the characteristic ordinary differential equations along the artificial boundary of the moving computational domain. The usage of this moving characteristic boundary conditions allows us to solve the HJ equations in any initial finite domain that we are interested in, regardless of the magnitude of the initial condition at the artificial domain boundary. Numerical tests in one and two dimensions are given to demonstrate the flexibility and efficiency of our schemes on the moving meshes in solving both smooth problems and problems with corner singularities.