Computed tomography entails the reconstruction of a function from measurements of its line integrals. In this talk we explore the question: How many and which line integrals should be measured in order to achieve a desired resolution in the reconstructed image? Answering this question may help to reduce the amount of measurements and thereby the radiation dose, or to obtain a better image from the data one already has. Our exploration leads us to a mathematically and practically fruitful interaction of Shannon sampling theory and tomography. For example, sampling theory helps to identify efficient data acquisition schemes, provides a qualitative understanding of certain artifacts in tomographic images, and facilitates the error analysis of some reconstruction algorithms. On the other hand, applications in tomography have stimulated new research in sampling theory, e.g., on nonuniform sampling theorems and estimates for the aliasing error. Our dual aim is an exposition of the main principles involved as well as the presentation of some new insights.