When an X-ray wave traverses an object, its amplitude and phase change, resulting in attenuation, interference, and refraction, and in phase-contrast X-ray imaging (PCI) these are converted to intensity changes. The relative change of the X-ray phase per unit path length is even orders of magnitude larger than that of the X-ray amplitude, so that the image contrast based on variation of the X-ray phase is potentially much stronger than the contrast based on X-ray amplitude (absorption contrast). An important medical application of PCI methods is soft-tissue imaging, where the absorption contrast is inherently weak. It is shown by in vitro examples that signs of malignant human breast tumor are enhanced in PCI images. Owing to the strong contrast, the radiation dose can be greatly reduced, so that a high-resolution phase-contrast X-ray tomography of the breast is possible with about 1 mGy mean glandular dose. Scattered radiation carries essential information on the atomic and molecular structure of the object, and particularly small-angle X-ray scattering can be used to trace cancer. The imaging methods developed at the synchrotron radiation facilities will become available in the clinical environment with the ongoing development of compact radiation sources, which produce intense X-ray beams of sufficient coherence. Several developments that are under way are described here.