Fluid-Structure Interaction of thin, flexible structures is omnipresent in nature and engineering. Numerical simulation of this kind of system is challenging due to the large non-linear deformations and the strong added-mass effect. In this thesis, we develop a novel immersed boundary fluid-structure structure interaction solver to deal with those challenging cases. First, we show that immersed boundary methods must explicitly impose the Neumann condition on the pressure field for accurate results when the structure is thin and dynamic. We develop an extension to an existing immersed boundary method, BDIM-sigma, that enforces this boundary condition regardless of the body’s thickness. The method relies on a variable coefficient Poisson equation to impose the Neumann condition on the interface. Standard linear algebra methods solve the resulting linear system. The method drastically outperforms Direct-Forcing methods and the standard immersed boundary method for problems with thin dynamic structures. We couple our immersed boundary method with a three-dimensional shell solver via an implicit partitioned approach. A quasi-Newton method solves the resulting fixed-point problem. The method constructs an approximation of the Jacobian of the interface via input-output pairs of the coupling variables from the previously converged time steps. These pairs form an inverse least-square problem whose solution is the (inverse) Jacobian of the interface. With various fluid-structure interaction examples, we show that the method possesses excellent spatial convergence properties, is stable and efficient for a large range of flexibility and mass ratios and outperforms standard Gauss-Seidel relaxation methods.Finally, we demonstrate the capability of our coupled solver to deal with practical examples by simulating bat flight. We investigate the effects of Strouhal number, membrane elasticity and fibre reinforcement on the aerodynamic efficiencies of bats. We show that the three-dimensional nature of the kinematics results in aerodynamic efficiencies peaking well outside the optimal Strouhal number range. Additionally, we show that propulsive efficiency is well correlated with membrane elasticity, where elastic membranes outperform stiff ones until flutter occurs. To extend the flutter envelope of the wing, we reinforce the isotropic membrane with anisotropic fibres. The response of this modified membrane shows a drastic reduction in the flutter and large improvements in aerodynamic efficiencies.