A new kind of tomographic X-ray imaging modality is discussed, where the patient is radiated as little as possible while recovering enough three-dimensional information for the clinical task at hand. The input can be only a dozen projection images or so, collected from different directions. Such sparse data typically represent limited-angle and local tomography configurations and lead to severely ill-posed reconstruction problems. This differs from traditional CT imaging, where a comprehensive data set is collected and the (only mildly ill-posed) reconstruction problem is solved using the classical filtered back-projection (FBP) algorithm. The incompleteness of sparse data violates the assumptions of FBP, leading to unacceptable reconstruction quality. However, statistical inversion methods can be used with sparse tomographic data. They yield clinically useful reconstructions, as demonstrated by real-data examples related to mammography, surgical imaging and dental imaging. Some of these methods have already entered commercial products: see http://www.vtcube.com.