Unfitted/Immersed or Embedded Finite Element methods have gained significant attention in the past decade for their effectiveness in simulating problems in complex geometries. However, many existing methods in the literature necessitate ad-hoc modifications of standard Finite Element data structures. This often involves tessellation of elements intersected by the embedded boundary, the construction of special quadrature rules, or the definition of geometry-dependent surrogate boundaries. In this work, we propose a novel approach: a generalization of the Weighted Shifted Boundary method [1], which eliminates the need for special data structures or quadrature rules, and enables the solution of unfitted problems with geometry-independent Finite Element spaces. This newly introduced framework holds particular significance for problems involving evolving geometries and potential topological changes. Applications extend to scenarios such as topology optimization or fluid-structure interaction problems where the ability to handle dynamic geometries is crucial. In this talk, we will present the formulation of the Generalized Weighted Shifted Boundary method and showcase its application to a variety of Fluid-Structure Interaction problems. [1] O. Colomés, A. Main, L. Nouveau and G. Scovazzi (2021). A weighted shifted boundary method for free surface flow problems. Journal of Computational Physics, 424, 109837.