We review various issues related to the direct detection of constituents of dark matter, which are assumed to be Weakly Interacting Massive Particles (WIMPs). We specifically consider heavy WIMPs such as: 1) The lightest supersymmetric particle LSP or neutralino.2) The lightest Kaluza-Klein particles in theories of extra dimensions and 3) other extensions of the standard model. In order to get the event rates one needs information about the structure of the nucleon as well as as the structure of the nucleus and the WIMP velocity distribution. These are also examined Since the expected event rates for detecting the recoiling nucleus are extremely low and the signal does not have a characteristic signature to discriminate them against background we consider some additional aspects of the WIMP nucleus interaction, such as the periodic behavior of the rates due to the motion of Earth (modulation effect). Since, unfortunately, this is characterized by a small amplitude we consider other options such as directional experiments, which measure not only the energy of the recoiling nuclei but their direction as well. In these, albeit hard, experiments one can exploit two very characteristic signatures: a)large asymmetries and b) interesting modulation patterns. Furthermore we extended our study to include evaluation of the rates for other than recoil searches such as: i) Transitions to excited states, ii) Detection of recoiling electrons produced during the neutralino-nucleus interaction and iii) Observation of hard X-rays following the de-excitation of the ionized atom.