The CRESST experiment, located at Laboratori Nazionali del Gran Sasso in Italy, searches for dark matter particles via their elastic scattering off nuclei in a target material. The CRESST target consists of scintillating CaWO$_4$ crystals, which are operated as cryogenic calorimeters at millikelvin temperatures. Each interaction in the CaWO$_4$ target crystal produces a phonon signal and a light signal that is measured by a second cryogenic calorimeter. Since the CRESST-II result in 2015, the experiment is leading the field of direct dark matter search for dark matter masses below 1.7,GeV/$c^2$, extending the reach of direct searches to the sub-GeV/$c^2$ mass region. For CRESST-III, whose Phase 1 started in July 2016, detectors have been optimized to reach the performance required to further probe the low-mass region with unprecedented sensitivity. In this contribution the achievements of the CRESST-III detectors will be discussed together with preliminary results and perspectives of Phase 1.
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