This review of different principles used in x-ray computerized tomography (CT) starts with attenuation (transmission) CT. The pros and cons of different geometrical solutions, single-ray, fan-beam and cone-beam, are discussed. Attenuation CT measures the spatial distribution of the linear attenuation coefficient, . The contributions of different interaction processes to have also been used for CT. Fluorescence CT is based on measurements of the contribution, , from an element Z with concentration , to the linear attenuation coefficient. Diffraction CT measures the differential coherent cross section , Compton CT the incoherent scatter cross section . The usefulness of these modalities is illustrated. CT methods based on secondary photons have a competitor in selected volume tomography. These two tomography methods are compared. A proposal to perform Compton profile tomography is also discussed, as is the promising method of phase-contrast x-ray CT.