Unfitted Finite Element (FE) methods are attractive for Computational Fluid Dynamics (CFD) and Fluid-Structure interaction (FSI) problems dealing with complex geometries. Several unfitted FE methods have been developed during the last decade, preserving accuracy and well-posedness of the system. Nonetheless, the definition of robust unfitted FE methods for high-order FE spaces is still an open topic. In a recent contribution we introduced the interpolation-based AgFEM [1], which is robust for high-order FEs. This is very relevant for CFD and FSI problems, as high-order FEs result in more accurate solutions for a given number of degrees of freedom, see for instance [2]. In this talk we will present the extension of the AgFEM method to FSI problems involving elastic bodies with complex geometries. Here we combine the high-order interpolation-based AgFEM method with an Arbitrary Lagrangian-Eulerian (ALE) approach that tracks the solid deformation in time. This approach enables the use of inf-sup stable velocity-pressure pairs of high-order, with an efficient integration on the cut cells. The proposed approach results in an accurate, optimal and well-conditioned system. We will demonstrate the suitability of this approach for a variety of tests including laminar and turbulent flows around elastic objects with complex geometries. REFERENCES [1] S. Badia, E. Neiva and F. Verdugo. Robust high-order unfitted finite elements by interpolation-based discrete extension. Computers & Mathematics with Applications, 127 (2022): 105-126. [2] O. Colom´es and S. Badia. Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows. Computer Methods in Applied Mechanics and Engineering, 313 (2017): 189-215.