The CRESST experiment is set up for the direct detection of Weakly Interacting Massive Particles (WIMPs) which our Galactic dark matter halo possibly consists of. the employed detection method is elastic scattering by nuclei. the recoiling nucleus deposits most of its energy in the form of lattice vibrations in the detector. Cooling the detector to very low temperatures (mK) enhances the temperature rise due to the energy deposition. the crucial parameter for direct WIMP searches is the sensitivity to the WIMP interaction cross section in a certain range of possible WIMP masses. CRESST is now sensitive enough to explore the parameter space predicted by supersymmetric models.