The AREPO code is a cosmological hydrodynamical simulation code on a fully dynamic unstructured mesh. At present, hydrodynamic cosmological simulations usually employ either the Lagrangian smoothed particle hydrodynamics (SPH) technique, or Eulerian hydrodynamics on a Cartesian mesh with (optional) adaptive mesh refinement (AMR). Both of these methods have disadvantages that negatively impact their accuracy in certain situations, for example the suppression of fluid instabilities in the case of SPH, and the lack of Galilean-invariance, the presence of overmixing and preferred directions in the case of AMR. This makes it highly desirable to develop novel numerical schemes that are more accurate in the demanding regime relevant for cosmic structure formation. Here, large advection velocities and supersonic flows are the norm, the density contrast is highly time-variable and reaches extreme values.