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Assessing and understanding the performance of photon-counting X-ray detectors is an essential prerequisite for successfully applying them to Spectral Computed Tomography. After a short overview of the technology, the detector response function is introduced as a characteristic spectral property of photon-counting detectors from which many performance parameters can be extracted. Subsequently we discuss how different physical effects like charge sharing between pixels or pulse pile-up at high photon fluxes affect this detector response function. In a second part several figures of merit are presented for a quantitative comparison of photon-counting detectors. The definition, measurement, interpretation, and limitations of quantities like the energy resolution, quantum, and spectral efficiency are discussed. The last part reviews different approaches to improve the spectral response of photon-counting detectors. Fast pulse shaping, charge-sharing and pile-up rejection, or charge summing mechanisms help to clean the measured spectrum. We analyze their advantages and disadvantages with respect to improving the detector response function. It is the hope of the authors that the reader will gain a good understanding about the assessment of the spectral performance of photon-counting detectors as well as about the current state of the technology.