Fluid--structure interaction (FSI) is a particularly important subclass of multi-physics problems that arise frequently in applications such as, e.g., wind turbines, naval architecture or biomedical applications. Performing shape optimization on this class of problems is also a vivid area of research. In this talk, we consider the method of mappings for performing shape optimization for unsteady FSI problems. Similar to the monolithic arbitrary Lagrangian--Eulerian (ALE) technique that we use to tackle the FSI problem, we introduce a fixed reference domain and represent shapes via domain transformations. We use a perturbation of identity ansatz and represent transformations via a displacement field. This leads to an optimization problem with PDE constraints for which the displacement field serves as control. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. To do so, we parameterize the displacements by a scalar valued function on the design boundary and introduce sophisticatedly chosen boundary and extension operators. We discretize the optimization problem such that we compute exact discrete gradients. This allows to use efficient tools and optimization algorithms that need not be tailor-made for shape optimization. Our focus is on problems derived from an FSI benchmark in order to validate our numerical implementation using FEniCS, dolfin-adjoint and IPOPT. Moreover, we present numerical results for optimizing parts of the outer boundary and the interface.