In the numerical simulation for fluid-structure problems, many different possible combinations of coordinate systems can be used for fluid and structure problems.   While Eulerian coordinate is almost always used for fluid problems,  either Eulerian or Lagrangian coordinate can be used for structure problems.   In this talk, I will discuss some mathematical and numerical issues for these different choices of coordinate systems.  In particular, I will report some convergence analysis for eXtended Finite Element Methods (XFEM) for Eulerian-Eulerian coordinates and the study of the well-posedness and preconditioning for finite element discretization based Arbitrary Lagrangian-Elerian (ALE) coordinates.  I will also present some numerical results for some practical applications such as hydroelectric power generator and abdominal aortic aneurysm.