At present, hydrodynamical simulations in computational star formation are either carried out with Eulerian mesh-based approaches or with the Lagrangian smoothed particle hydrodynamics (SPH) technique. Both methods differ in their strengths and weaknesses, as well as in their error properties. It would be highly desirable to find an intermediate discretization scheme that combines the accuracy advantage of mesh-based methods with the automatic adaptivity and Galilean invariance of SPH. Here we briefly describe the novel AREPO code which achieves these goals based on a moving unstructured mesh defined by the Voronoi tessellation of a set of discrete points. The mesh is used to solve the hyperbolic conservation laws of ideal hydrodynamics with a finite volume approach, based on a second-order unsplit Godunov scheme with an exact Riemann solver. A particularly powerful feature is that the mesh-generating points can in principle be moved arbitrarily. If they are given the velocity of the local flow, an accurate Lagrangian formulation of continuum hydrodynamics is obtained that features a very low numerical diffusivity and is free of mesh distortion problems. If the points are kept fixed, the scheme is equivalent to a Eulerian code on a structured mesh. The new AREPO code appears especially well suited for problems such as gravitational fragmentation or compressible turbulence.