We are now developing a system which allows the visualization of the structure and tissue associated with breast cancer, which is difficult to achieve with conventional mammography, by utilizing X-ray refraction. This system is expected to achieve an approximately 1000-fold higher contrast and high spatial resolution at low dose; compared to techniques by use of X-ray absorption. Monochromatic and plane wave X-rays, produced by the diffraction of asymmetric reflection, are applied to the sample, and, corresponding to calcification and tissue structure, X-rays, which is going through slight refraction, are isolated from straight X-rays with an angle analysis plate. With a method called the 'X-ray dark field method' which uses a tramsmission type angle analysis plate, transfer imaging created by refractive X-rays alone can be obtained, making use of the feature that straight X-rays do not go through the angle analysis plate of a certain thickness. Furthermore, a mathematical algorithm concerning refraction contrast was developed to get CT images, and we obtained world's first 3D images of noninvasive ductual carcinoma, etc., based on the refraction principle.