I will present several models describing the interactions between viscous incompressible fluids and solids, which may be either rigid or deformable. The governing equations for these interactions consist of the Navier Stokes equations coupled with either the Euler equations for rigid body dynamics or the Navier equations of linearized elasticity. A common feature of these problems is that the governing equations have a dissipative conservative (parabolic hyperbolic) structure: while the fluid introduces dissipation in the mechanical energy of the coupled system, the solid contributes to the same energy with a conservative (more generally, non-decreasing) component. From a mathematical perspective, this interplay poses significant challenges when establishing the existence and stability of solutions to the governing equations. I will provide an overview of my contributions to the analysis of these problems for various mechanical systems involving fluid solid interactions.