The neutralino is considered as a particle of dark galactic matter. So
me general problems encountered in direct searches for dark matter by
detecting events of elastic neutralino-nucleus scattering are discusse
d. On the basis of the most general ideas of nuclear and nucleon struc
tures, constraints that direct detection of dark matter may impose on
the parameter space of supersymmetric models are analyzed. A criterion
of sensitivity to various groups of parameters that characterize supe
rsymmetric models is proposed. According to this criterion, all detect
ors of dark matter are divided into three categories. On the basis of
the minimal supersymmetric standard model (MSSM), the event rate for n
eutralino-nucleus scattering is calculated for various isotopes that a
re of interest for developing dark-matter detectors. In this calculati
on, accelerator and cosmological constraints on the MSSM parameter spa
ce are taken into account. The effect of nuclear spin on elastic neutr
alino-nucleus scattering is studied. It is found that the role of spin
-dependent interaction in neutralino-nucleus scattering is of minor im
portance for nuclei with mass numbers that are not less than 50.