Phase-contrast x-ray computed tomography (CT) is an emerging imaging technique that can be implemented at third generation synchrotron radiation sources or by using a microfocus x-ray tube. Promising experimental results have recently been obtained in material science and biological applications. At the same time, the lack of a mathematical theory comparable to that of conventional absorption-based CT limits the progress in this field. We suggest such a theory and prove a fundamental theorem that plays the same role for phase-contrast CT as the Fourier slice theorem does for absorption-based CT. The fundamental theorem allows us to derive fast image reconstruction algorithms in the form of filtered backprojection (FBP).© (2006) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.