Interaction problems between a viscous incompressible fluid and a (rigid or deformable) solid are widely studied because of their applications in hemodynamics, geophysics and engineering. The mathematical models governing this type of problems feature a dissipative component (typically through the Navier–Stokes equations for the fluid part) and a conservative component (due to the solid counterpart, through either the classical equations of rigid body dynamics or the equations of elasticity). This dissipative-conservative interplay affects the global dynamics of the fluid-solid systems, and it produces mathematical challenges when studying fundamental questions like existence, uniqueness, stability and regularity of solutions to the relevant equations of motion. I will present different mechanical systems featuring fluid-solid interactions, and whose governing equations are characterized by the above-mentioned dissipative-conservative interplay. Open questions and current research interests will be also discussed.