There is a pressing need for a comprehensive theory for phase-contrast x-ray imaging to guide its development and clinical applications. This work presents such a theory as the foundation for deriving these guidelines. The new theory is based on the Wigner-distributions for the parabolic wave equations, and it is more general than the present theories based on the Fresnel-Kirchhoff diffraction theory. The new theory shows for the first time how the complex degree of coherence (CDC) of the incident x-ray beam determines the phase-contrast visibility in general, and how the reduced complex degree of coherence (RCDC) for an anode-source is equal to the system's optical transfer function for geometric unsharpness in particular. The role of detector resolution in phase visibility has been clarified as well. Computer simulations based on the new theory were conducted and optimal design parameters were derived for phase-contrast mammography systems. © 2004 American Association of Physicists in Medicine.