This paper presents a new Arbitrary Lagrangiae-Eulerian (ALE) finite element method for the analysis of three-dimensional large-amplitude sloshing problems. The present method is composed of three phases, i.e., Phase 1: Purely Lagrangian calculation. Phase 2: Remeshing procedure and Phase 3: Rezoning procedure. By applying these procedures, the present method has succeeded in overcoming the main difficulty of computation in the large-amplitude sloshing analysis. In the Lagrengian calculation, a new fractional step method that satisfies the free-surface boundary condition has been applied, and treatment of the free surface on the multi-sloped wall boundary has been advanced. In the remeshing procedure, a method based on boundary-fitted coordinates is introduced. Numerical results are compared with experimental results, and it is shown that this method is sufficiently accurate to calculate the three-dimensional sloshing. Furthermore, a numerical example of large-amplitude sloshing in a tank with roofs and chamfers illustrates the validity of the method for the computation of large-amplitude sloshing waves whose free surface configuration is a double-valued function.