  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.
