In this talk, we will present new families of high order one-step fully direct explicit schemes of Finite Volume (FV) and Discontinuous Galerkin (DG) type for the solution of hyperbolic systems. In the first part, we will concentrate on the novelties we introduced in the framework of direct Arbitrary Lagrangian-Eulerian (ALE) schemes by using moving Voronoi tessellations that, at each time step, are regenerated and are even free to change their topology. The new coupling, between arbitrary high order accurate direct schemes and high quality moving meshes, is achieved here, for the first time in literature, by integrating over arbitrary shaped closed space--time control volumes (including degenerated space time elements and sliver elements) a fully-discrete space--time conservation formulation of the governing hyperbolic PDE system. The obtained scheme is conservative and automatically satisfies the geometric conservation law (GCL) by construction. In the second part, we will consider a covariant formulation (i.e. independent of the coordinate system) of the General Relativistic Hydrodynamics (GRHD) model, studied at the aid of our Well Balanced Path-Conservative techniques for nonconservative systems that allow to preserve stationary solutions exactly at the discrete level and so to enhance the power of resolution of our schemes. Future research aims at enhancing the two algorithms and coupling them together.