In this talk, I consider the existence of local-in-time strong solutions to a well established coupled system of partial differential equations arising in Fluid-Structure interactions. The system consisting of an incompressible Navier-Stokes equation and an elasticity equation with velocity and stress matching boundary conditions at the interface in between the two domains where each of the two equations is defined. I discuss new existence results for a range of regularity in the initial data and the differences in the exsitence results when domains with non-flat boundaries are considered.