We discuss the short and long term perspectives of the CRESST (cryogenic rare event search using superconducting thermometers) project and present the current status of the experiment and new results concerning detector development. In the search for elementary particle dark matter, CRESST is presently the most advanced deep underground, low background, cryogenic facility able to house up to about 100 kg of target mass. The basic technique involved is to search for WIMPS (weakly interacting massive particles) by the measurement of non-thermal phonons, as created by WIMP-induced nuclear recoils. When this technique is combined with our newly developed method for the simultaneous measurement of scintillation light, strong background discrimination is possible, resulting in a substantial increase in WIMP detection sensitivity. In contrast to other projects, CRESST technology allows the employment of a large variety of detection materials. This offers a powerful tool in establishing a WIMP signal and in investigating WIMP properties in the event of a positive signal. In addition it is useful in the understanding of background. Short term plans include the continuation of our current measurements with sapphire crystals for the detection of light WIMPs. The new scintillating detectors, based on CaWO_4, will be especially sensitive to medium and high mass WIMPs, and will allow a test of the reported positive evidence for a WIMP signal by the DAMA collaboration in the near future. In the long term, the present CRESST set-up permits the installation of a detector mass up to 100 kg and so can compete favourably with other large volume dark matter search proposals. (orig.)