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We describe recent progress on the use of Schottky CdTe diode detectors for spectrometry. The low leakage current of the CdTe diode allows us to apply a much higher bias voltage than was possible with previous CdTe detectors. For a relatively thin detector of 0.5-1 mm thickness, the high bias voltage results in a high electric field in the device. Both the improved charge-collection efficiency and the low-leakage current lead to an energy resolution of better than 600 eV full-width at half-maximum at 60 keV for a 2×2 mm2 device without any charge-loss correction electronics. Large-area detectors with dimensions of 21×21 mm2 are now available with an energy resolution of ∼2.8 keV. Long-term stability can be easily attained for relatively thin (<1 mm) detectors if they are cooled or operated under a high bias voltage.