We introduce and analyze a high-order Discontinuous Galerkin method on polytopal meshes (PolyDG) to discretize fluid-structure interaction problems [1,2]. In particular, we propose a method where the fluid mesh (on the background) is fixed, apart from the interface with the moving immersed structure, where general polytopal elements of arbitrary shape and changing in time are generated. The main features of the proposed method are: it can handle non-conforming meshes at the interface; it can naturally support polytopal fluid mesh elements arising at the interface as a consequence of the cut; it naturally accommodates high-order discretizations, by using modal basis functions built on the physical mesh element. In the computational practice, for high-order and/or three-dimensional computations numerical integration should be treated with care, as the classical sub-tessellation method becomes unfeasible due to CPU time burden. In such cases, the quadrature-free method proposed in [3] can be successfully employed. We prove a stability result of the proposed semi-discrete formulation and discuss how to deal with the partial or total covering of a fluid mesh element due to the structure movement. We finally present some numerical results to show the effectiveness of the proposed method. We also discuss how to further develop suitable contact algorithms for two or more immersed structures, based again on the PolyDG approach. [1] S. Zonca, P.F. Antonietti, C. Vergara. A Polygonal Discontinuous Galerkin formulation for contact mechanics in fluid-structure interaction problems. Commun. Computat. Phys., 30:1-33, 2021 [2] P. Antonietti, M. Verani, C. Vergara, and S. Zonca. Numerical solution of fluid-structure interaction problems by means of a high order Discontinuous Galerkin method on polygonal grids. Finite Elem. Anal. Des., 159:1–14, 2019 [3] P. F. Antonietti, P. Houston, and G. Pennesi. Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods. J. Sci. Comput., 77(3):1339–1370, 2018