In this talk we present a novel high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Discontinuous Galerkin (DG) scheme with a posteriori subcell Finite Volume (FV) limiter, on moving Voronoi meshes that, at each time step, are regenerated, thus connectivity and topology changes may occur. The Voronoi tessellation is obtained from a set of generator points moving according to a high order approximation of the local fluid velocity, combined with suitable mesh optimization techniques, that both maintain a high quality of the moving mesh and, thanks to a deferred-in- time approach to connectivity changes, reduce the occurrence of topology changes around shock and contact waves. Then, the old and new elements associated to the same generator point are connected in space and time to construct the so-called space-time control volumes, whose bottom and top faces may be different polygons; also degenerate sliver elements are incorporated in order to fill the space–time holes that arise due to the topology changes. The final ALE DG scheme is obtained by integrating a space-time conservation formulation of the governing hyperbolic PDE system over the Voronoi and sliver space–time control volumes. The obtained scheme satisfies the GCL by construction and is conservative thanks to the careful treatment of the space–time holes. In order to give clear numerical evidences that the novel deferred mesh optimization greatly improves the robustness of the scheme without affecting its Lagrangian character, we show a set of simple and very salient numerical results. In addition, we will briefly present a quasi- conservative approach able to capture contacts avoiding any spurious numerical artefacts, thanks to the PDE evolution in primitive variables, and at the same time strongly conservative on shocks, thanks to the employed a posteriori conservative limiter. Applications to Euler equations for single and multimaterial flows (on fixed meshes) will be shown.