In this work a structured adaptive mesh refinement (S-AMR) strategy for fluid-structure interaction (FSI) problems in laminar and turbulent incompressible flows is developed. The Eulerian computational grid consists of nested grid blocks at different refinement levels. The grid topology and data-structure is managed by using the Paramesh© toolkit. The filtered Navier-Stokes equations are evolved in time by means of an explicit second-order projection scheme, where spatial derivatives are approximated with second order central differences on a staggered grid. The level of accuracy of the required variable interpolation operators is studied, and a novel divergence-preserving prolongation scheme for velocities is evolved. A novel direct-forcing embedded-boundary method is developed to enforce boundary conditions on a complex moving body not aligned with the grid lines. In this method, the imposition of no-slip conditions on immersed bodies is done on the Lagrangian markers that represent their wet surfaces, and the resulting force is transferred to the surrounding Eulerian grid points by a moving least squares formulation. Extensive testing and validation of the resulting strategy is done on a numerous set of problems. For transitional and turbulent flow regimes the large-eddy simulation (LES) approach is used. The grid discontinuities introduced in AMR methods lead to numerical errors in LES, especially if non-dissipative, centered schemes are used. A simple strategy is developed to vary the filter size for filtered variables around grid discontinuities. A strategy based on explicit filtering of the advective term is chosen to effectively reduce the numerical errors across refinement jumps. For all the FSI problems reported, the complete set of equations governing the dynamics of the flow and the structure are simultaneously advanced in time by using a predictor-corrector strategy. Dynamic fluid grid adaptation is implemented to reduce the number of grid points and computation costs. Applications to flapping flight comprise the study of flexibility effects on the aerodynamic performance of a hovering airfoil, and simulation of the flow around an insect model under prescribed kinematics and free longitudinal flight. In the airfoil simulations, it is found that peak performance is located in structural flexibility-inertia regions where non-linear resonances are present.