A fully coupled numerical methodology is developed for calculating the flow-structure inter- action problems. The Roe scheme is extended to moving grid and used with the finite-volume method. The unsteady solutions march in time by using a dual-time stepping implicit unfac- tored line Gauss-Seidel iteration. The unsteady Navier-Stokes equations and the linear struc- tural equations are fully coupled implicitly via successive iteration with pseudo time stepping. The moving mesh and mesh deformation strategy is based on two mesh zones, a fine mesh zone surrounding the solid body without mesh deformation and a coarse mesh zone surrounding the fine mesh zone and deforms with the solid object. This mesh deformation strategy can maintain the orthogonality of the mesh near the wall and save CPU time for re-meshing. The study cases presented include a vortex-induced oscillating cylinder, a forced pitching airfoil, and an elastically mounted transonic airfoil. For the elastic transonic airfoil, the flutter boundary is calculated. Other phenomena captured include the limit cycle oscillation (LCO) and the steady state flow conditions, under which the aerodynamic forces and moments are balanced by the structure. The computational results agree well with the experiments and the computed results of other researchers. The methodology is demonstrated to be accurate, robust and effcient.