Exotic dark matter together with the vacuum energy (cosmological constant) seem to dominate in the flat Universe. Thus direct dark matter detection is central to particle physics and cosmology. Supersymmetry provides a natural dark matter candidate, the lightest supersymmetric particle (LSP). It is possible to obtain detectable rates, but realystically they are expected to be much lower than the present experimental goals. So one should exploit two characteristic signatures of the reaction, namely the modulation effect and the correlation with the sun's motion in directional experiments. In standard non directional experiments the modulation is small, less than two per cent. In the case of the directional experiments, the main subject of this paper, we find two novel features, which are essentially independent of the SUSY model employed, namely: 1) The forwardbackward asymmetry, with respect to the sun's direction of motion, is very large and 2) The modulation observed in a plane perpendicular to the sun's motion can be higher than 20 per cent and is direction dependent.