It has been proven by several astronomical observations that dark matter exists, but no particle
candidates have been observed yet. The CRESST experiment aims to directly detect dark matter
particles elastically scattering off nuclei in CaWO4 crystals which are operated at mK temperatures.
With nuclear recoil energy thresholds as low as 0.3 keV [2] and 0.6 keV [3], for the detector
modules LISE and TUM40, respectively, CRESST is ideally suited for the detection of low-mass
dark matter particles [5]. Additionally, the radiopurity of the crystals is another important factor
for the detector performance. For a detailed understanding of the detector backgrounds, we simulate
the radioactive contaminations of the TUM40 detector module with Geant4. The outcome of
this simulation will be vital for the CRESST-III experiment. In this contribution, we discuss our
results of the search for dark matter and dark photons achieved with the detector module Lise of
CRESST-II. We will discuss the status of CRESST-III Phase 1 which started taking data in 2016.