Photon counting X-ray detectors (PCDs) with spectral capabilities have the potential to revolutionize computed tomography (CT) for medical imaging. The ideal PCD provides accurate energy information for each incident X-ray, and at high spatial resolution. This information enables material-specific imaging, enhanced radiation dose efficiency, and improved spatial resolution in CT images. In practice, PCDs are affected by nonidealities, including limited energy resolution, pulse pileup, and cross talk due to charge sharing, K-fluorescence, and Compton scattering. In order to maximize their performance, PCDs must be carefully designed to reduce these effects and then later account for them during correction and post-acquisition steps. This review article examines algorithms for using PCDs in spectral CT applications, including how nonidealities impact image quality. Performance assessment metrics that account for spatial resolution and noise such as the detective quantum efficiency (DQE) can be used to compare different PCD designs, as well as compare PCDs with conventional energy integrating detectors (EIDs). These methods play an important role in enhancing spectral CT images and assessing the overall performance of PCDs.