Nowadays, fluid-structure interaction problems are a great challenge of different fields in engineering and applied sciences. Wind flow and structural motion may lead to aeroelastic instabilities on constructions such as long-span bridges, high-rise buildings and light-weight roof structures. Biomechanical applications are interested in the study of blood flow through large arteries, where membrane deformations interact with incompressible fluids. In the structural part of this work, a new methodology for the analysis of geometrically nonlinear orthotropic membrane and rotation-free shell elements is developed. The fluid portion of this work is governed by the incompressible Navier-Stokes equations, which are modeled by stabilized equal-order interpolation finite elements. The fluid-structure interaction problem is formulated as a three-field system: the structure, the fluid and the moving fluid mesh solver. Motion of the fluid domain is accounted for with the ALE formulation. Several examples are presented to verify the robustness and efficiency of the overall algorithm. This book is intended for engineers, scientists and postgraduates students interested in FSI.