The CRESST experiment (Cryogenic Rare Even Search with Superconducting Thermometers), 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 CaWO4 crystals, which are operated as cryogenic calorimeters at millikelvin temperatures. Each interaction in the CaWO4 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/c2, extending the reach of direct searches to the sub-GeV/c2 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.