We propose a general variational framework for the adaptive finite element approximation of fluid-structure interaction problems. The modeling is based on an Eulerian description of the (incompressible) fluid as well as the (elastic) structure dynamics. This is achieved by tracking the movement of the initial positions of all `material' points. In this approach the deformation appears as a primary variable in an Eulerian framework. Our approach uses a technique which is similar to the 'Level Set' method in so far that it also tracks initial data, in our case the set of 'Initial Positions', and from this determines to which `phase' a point belongs. To avoid the need for reinitialization of the initial position set, we employ the harmonic continuation of the structure velocity field into the fluid domain. Based on this monolithic model of the fluid-structure interaction we apply the 'dual weighted residual method' for goal-oriented a posteriori error estimation and mesh adaptation to fluid-structure interaction problems. Results from nonstationary examples are presented.