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. These ad-hoc modifications often prevent the use of these approaches for problems where the geometry is a varying parameter of the PDE at hand. In this work, we propose a novel approach: The Generalized 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 addition, the proposed methodology can also be relevant for machine-learning based surrogate models where the geometry is a parameter. In this talk, we will present the formulation of the Generalized Shifted Boundary method and showcase its application to a variety of problems with evolving domains. REFERENCES [1] O. Colomés, J. Modderman and G. Scovazzi, The Generalized Shifted Boundary Method for geometry-parametric PDEs and time-dependent domains, Computer Methods in Applied Mechanics and Engineering, In Press, 2026.