Problems involving fluid-structure interactions arise in different areas of application, traditionally in aero- elasticity, but also in bio-medical research. We focus on hemodynamical applications, in particular on the interaction of the blood-flow with the surrounding elastic walls of the vessels or the heart chamber. This type of application as two specific difficulties: the coupling between fluid and solid is very stiff due to the similar densities of both materials (about 103 kg=m3). This gives rise to stability problems known as the added-mass effect. Further, one has to deal with large deformations up to topology changes, when the structure gets into contact with other parts of the structure (closing heart valves). While the first difficulty asks for strongly coupled or monolithic models and solution schemes, the second problem rules out the use of the most established monolithic modeling strategy, the Arbitrary Lagrangian Eulerian (ALE) coordinates. Here, the flow-problems is mapped onto a fixed reference domain that always matches the reference-configuration of the structure. If however the topology is changing such a mapping cannot be differentiable or invertible and the resulting scheme will fail. In this talk we propose a novel monolithic model for fluid-structure interactions, where both systems, fluid and solid are given in a Eulerian formulation. By this construction we circumvent the use of an artificial fluid-domain mapping. Large deformation, movement or even contact are possible. This formulation however brings all difficulties of fixed-mesh methods along: the interface between fluid and solid is moving and must be captured with high accuracy. Since the domains of influence are changing with time, repeated projections of discrete solution to new meshes are required. We will discuss the prospects of this Fully Eulerian approach, give remarks on the various technical difficulties and will demonstrate it’s potential to treat stiffly coupled fluid-structure applications with large deformation and contact. References: [1] T. Richter. A fully eulerian formulation for fluid-structure-interaction problems. J. Comp. Phys., 2012. published online. [2] T. Richter and T. Wick. Finite elements for fluid-structure interaction in ale and fully eulerian coordinates. Computer Methods in Applied Mechanics and Engineering, 199:2633–2642, 2010. [3] R. Rannacher and T. Richter. An adaptive finite element method for fluid-structure interaction problems based on a fully eulerian formulation. In H.J. Bungartz, M. Mehl, and M. Sch”afer, editors, Fluid-Structure Interaction II, Modelling, Simulation, Optimization, number 73 in Lecture notes in computational science and engineering, pages 159–192. Springer, 2010. [4] T. Dunne. An eulerian approach to fluid-structure interaction and goal-oriented mesh refinement. Int. J. Numer. Math. Fluids., 51:1017–1039, 2006.