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The energy spectrum of X-ray photons after passage through an absorber contains information about its elemental composition. Thus, tissue characterisation becomes feasible provided that absorption characteristics can be measured or differentiated. Dual-energy CT uses two X-ray spectra enabling material differentiation by analysing material-dependent photo-electric and Compton effects. Elemental concentrations can thereby be determined using three-material decomposition algorithms. In comparison to dual-energy CT used in clinical practice, recently developed energy-sensitive photon-counting detectors sample the material-specific attenuation curves at multiple energy levels and within narrow energy bands; the latter allows the detection of element-specific, k-edge discontinuities of the photo-electric cross section. Multi-energy CT imaging therefore is able to concurrently identify multiple materials with increased accuracy. These specific data on material distribution provide information beyond morphological CT, and approach functional imaging. This article reviews the principles of dual- and multi-energy CT imaging, hardware approaches and clinical applications.