In this talk I present some recent advances on a fictitious domain approach for the discretization of fluid-structure interaction problems proposed in [1]. This formulation is based on the introduction of a distributed Lagrange multiplier so that the problem fits in the framework of saddle point systems. Our formulation allows for solving the Navier-Stokes equation and the elasticity equation on meshes independent of each other, at the price of computing a coupling term which involves test funtions defined on both meshes. A discussion on how to deal carefully with such term will also be presented, see [2] and [3]. We shall show also that the formulation is robust with respect small cells. REFERENCES [1] D. Boffi and L. Gastaldi, “A fictitious domain approach with Lagrange multiplier for fluid-structure interactions”, Numer. Math., Vol. 135 pp.711-732 (2017). [2] D. Boffi, F. Credali and L. Gastaldi. “On the interface matrix for fluid-structure interaction problems with fictitious domain approach”, Comp. Meth. Appl. Mech. Eng. 401 (2022). [3] D. Boffi, F. Credali and L. Gastaldi. “Quadrature error estimates on non-matching grids in a fictitious domain framework for fluid-structure interaction problems”, submitted.