Unfitted finite element methods such as XFEM, CutFEM and Finite Cell are a popular modern tool for the numerical approximation of partial differential equations on complex geometries, which allows the separation of the geometry description from the computational mesh. Key to these approaches is stabilization of small cuts. It is desired to be able to combine this with methods that have additional conservation properties (e.g.~mass conservation), such as H(div)-conforming finite element methods for the mixed Poisson problem. The discrete problem needs to be modified and stabilized in the unfitted case. On the one hand this needs to be done to obtain a suitable Ladyzhenskaya–Babuska–Brezzi (LBB) condition and stability that accounts for arbitrarily small cuts. On the other hand, this needs be achieved without polluting the conservation properties of the method. In this talk we discuss recent results on the analysis of a new H(div)-conforming unfitted finite element method. The method is both robust in the cut configuration and preserves conservation properties. Here we achieve robustness without the need for ghost penalties or cell agglomeration for the scalar variable. The discussed method introduces an inconsistency for the scalar variable, but not the flux variable. This inconsistency can be dealt with in post-processing. For the resulting scheme we show optimal convergence rates for both variables and even superconvergence for the post-processed scalar variable.