  We consider the neutralino as a dominant Dark Matter particle in the galactic
halo and investigate some general issues of direct Dark Matter searches via
elastic neutralino-nucleus scattering. On the basis of conventional assumptions
about the nuclear and nucleon structure we analyse constraints on SUSY model
parameter space accessible by the direct Dark Matter searches. This analysis
shows that Dark Matter detectors fall into the THREE different categories with
respect to their sensitivity to different groups of the SUSY model parameters.
We calculate the event rate for various experimentally interesting isotopes
within the Minimal Supersymmetric Standard Model (MSSM) taking into account the
known accelerator and cosmological constraints on the MSSM parameter space. We
investigate the role of nuclear spin in elastic neutralino - nucleus
scattering. It is found that the contribution of the spin-dependent interaction
to this process is subdominant for nuclei with atomic weights A > 50.
