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, mu. The contributions of different interaction processes to mu have also been used for CT. Fluorescence CT is based on measurements of the contribution, cZtauZ/rho, from an element Z with concentration cZ, to the linear attenuation coefficient. Diffraction CT measures the differential coherent cross section d sigma (theta)(coh)/d omega, Compton CT the incoherent scatter cross section sigma. 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.