Within this work, we consider optimization settings for nonlinear, nonstationary fluid-structure interaction (FSI). The problem is formulated in a monolithic fashion using the arbitrary Lagrangian-Eulerian framework to set-up the fluid-structure forward problem. In the optimization approach, either optimal control or optimal design problems are treated. In the latter, the stiffness of the solid is estimated from given reference values. In the numerical solution, the optimization problem is solved with a gradient-based solution algorithm. The nonlinear subproblems of the FSI forward problem are solved with a Newton method including line search. Specifically, we will formally provide the backward-in-time running adjoint state used for gradient computations. Our algorithmic developments are demonstrated with some numerical examples as, for instance, extensions of the well-known fluid-structure benchmark settings and a flapping membrane test in a channel flow with elastic walls.