This thesis concerns the study of flexible flapping wings and the related numerical methods. It thus contains two distinct themes. The first contribution is the implementation of an efficient fluid-structure interaction algorithm that handles the interaction of an elastic solid undergoing large displacement with an incompressible fluid. The algorithm is based on the partitioned approach and allows state-of-the-art fluid and structural solvers to be used. Stabilization with artificial compressibility in the fluid continuity equation along with judicious algorithmic choices make the method suitable to be used with SIMPLE or PISO projection fluid solvers. The second contribution is the study of the effects of wing flexibility in flapping flight. The different regimes, namely inertia-driven and pressure-driven wing deformations are presented along with their effects on the topology of the flow and, eventually, on the performance of the flapping wing in propulsion regime. It is found that pressure-driven deformations can increase the thrust efficiency if a suitable amount of flexibility is used. Thrust increases are also observed when small pitching amplitude cases are considered. On the other hand, inertia-driven deformations generally deteriorate aerodynamic performances of flapping wings unless meticulous timing is respected, making them less practical. It is also shown that wing flexibility can act as a passive pitching mechanism while keeping decent thrust and efficiency. Lastly, a freely-moving flexible flapping wing model is presented. It is shown that the deviation motion found in natural flyers is a consequence of a feathering mechanism.