The recent WMAP data have confirmed that exotic dark matter together with the vacuum energy (cosmological constant) dominate in the flat Universe. Thus the direct dark matter search, consisting of detecting the recoiling nucleus, is central to particle physics and cosmology. Modern particle theories naturally provide viable cold dark matter candidates with masses in the GeV-TeV region. Supersymmetry provides the lightest supersymmetric particle (LSP), theories in extra dimensions the lightest Kaluza-Klein particle (LKP) etc. In such theories the expected rates are much lower than the present experimental goals. So one should exploit characteristic signatures of the reaction, such as the modulation effect and, in directional experiments, the correlation of the event rates with the sun's motion. In standard non directional experiments the modulation is small, less than two per cent and the location of the maximum depends on the unknown particle's mass. In directional experiments, in addition to the forward-backward asymmetry due to the sun's motion, one expects a larger modulation, which depends on the direction of observation. We study such effects both in the case of a light and a heavy target. Furthermore, since it now appears that the planned experiments will be partly directional, in the sense that they can only detect the line of the recoiling nucleus, but not the sense of direction on it, we study which of the above mentioned interesting features, if any, will persist in these less ambitious experiments.

22 LaTex pages, 28 figures