The development of finite element discretizations for partial differential equations (PDEs) on non-body-fitted meshes has attracted significant attention due to their flexibility in handling complex and evolving geometries. In these approaches, the PDE is posed on a physical domain, while the numerical discretization is carried out on a computational mesh that does not conform to the boundaries or interfaces of the physical domain. Such methods are particularly well suited for problems involving evolving domains, interfaces, or topological changes. However, this geometric mismatch introduces several analytical and numerical challenges, especially in the design of stable and accurate high-order methods. This minisymposium aims to bring together researchers working on unfitted and related discretization techniques, including fictitious domain methods, CutFEM, X-FEM, $\phi$-FEM, the Shifted Boundary Method, the Transfer Path Method, HHO methods, and the Immersed Boundary Method, among others. Particular emphasis will be placed on recent advances and open challenges in high-order unfitted discretizations, as well as on the analysis, implementation, and applications of such methods more broadly.