The Fully Eulerian method is a monolithic approach for fluid-structure interactions, where the two fields are coupled in a fully variational setting. The striking property of the Fully Eulerian model is its capability to cope with arbitrary large deformations and contact while keeping the coupling monolithic. We study the Fully Eulerian approach as it was introduced by Dunne and Rannacher in 2006. For the first time, this manuscript gives a thorough discussion of accurate numericalmethods for the spatial and temporal discretization. The characteristic difficulty in the Fully Eulerian model is the interface - the coupling boundary between fluid and solid - that, in a Eulerian description moves freely within the domain. On fixed Eulerian background meshes, standard finite elements cannot properly resolve the interface, where the solution might have kinks with limited regularity (for the velocity) or even discontinuities (pressure). As the interface moves with time and will change its position on a Eulerian background mesh, certain nodal points might change their affiliation fromone side of the interface to the otherwithin one time-step. Regularity is limited in the temporal direction and accuracy will be lost, if standard time-stepping schemes are used. Another problem arises due to the Eulerian description of the solid equations, which brings along convective terms. In particular close to the interface, robust and stable discretization is difficult and asks for special stabilization tools. Themain part of this chapter introduces and analyses novel and high-order discretization schemes for interface problemswithmoving interfaces. For the first time in a fully Eulerian setting, we are able to derive numerical schemes of second order in space and time and show reliable results for different numerical test-cases.