  We compute the neutralino direct detection rate in non-standard cosmological
scenarios where neutralinos account for the dark matter of the Universe.
Significant differences are found when such rates are compared with those
predicted by the standard cosmological model. For bino-like neutralinos, the
main feature is the presence of additional light ($m_\chi\lesssim 40\gev$) and
heavy ($m_\chi\gtrsim 600\gev$) neutralinos with detection rates within the
sensitivity of future dark matter experiments. For higgsino- and wino-like
neutralinos lighter than $m_\chi \sim 1\tev$, enhancements of more than two
orders of magnitude in the largest detection rates are observed. Thus, if dark
matter is made up of neutralinos, the prospects for their direct detection are
in general more promising than in the standard cosmology.
