We present the first magneto-hydrodynamical (MHD) simulations of pre-stellar collapse with the moving-mesh code AREPO. We follow the collapse of rotating and
turbulent cloud cores of up to two solar masses to central densities of $\sim 10^{20}$ cm$^{-3}$ with a
maximum resolution of $\sim 10^{-3}$ AU. The impact of the non-ideal MHD effects from Ohmic
dissipation, ambipolar diffusion, and the Hall effect on the final disk structure are
investigated. For idealized rotating cloud setups, we find good agreement with recent
results from grid-based and particle-based non-ideal MHD simulations. We showcase the
strong impact of turbulence in the initial conditions on the final structure of the accretion
disks and discuss implications for the formation of protostellar accretion disks in the
realistic environments of turbulent molecular clouds and cloud cores.