The Weakly Interacting Massive Particles (WIMPs) are among the main candidates for the relic Dark Matter (DM). The idea of the direct DM detection relies on elastic Spin-Dependent (SD) and Spin-Independent (SI) interaction of WIMPs with target nuclei. In this review formulas for the DM event rate calculations are collected. The importance of the SD WIMP-nucleus interaction for reliable DM detection is argued and the relevant to DM search spin nuclear structure functions are discussed. The effective low-energy minimal supersymmetric standard model (MSSM) is used for calculation of the DM cross sections, provided the lightest neutralino is the WIMP. It is shown that the absolute lower bound for the rate of direct DM detection is due to the SD WIMP-nucleon interaction, and a new-generation experiment aimed at detecting DM with sensitivity higher than 10^-5 event/(kg·day) should have a nonzero-spin target to avoid missing of the DM signal. The mixed spin-scalar couplings approach is argued. Prospects of DM experiments with high-spin Ge-73 are discussed in the mixed coupling scheme. The DAMA experiment has claimed observation of the WIMPs due to the annual signal modulation. Some important consequences of the DAMA claim for the other DM searches as well as for collider physics are considered.

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