No future X-ray telescope is likely ever to have angular resolution significantly superior to the Chandra X-Ray Observatory without adopting a new technology. We consider focusing systems based upon diffraction and refraction rather than grazing incidence reflection. The elements are Fresnel zone plates and refractive lenses in configurations where chromatic aberration is corrected over a finite bandwidth. This technique is likely to be especially effective in the intermediate regime between the 0.5 arcsec capability of the Chandra telescope and the one-tenth microarcsec resolution required for the NASA "Vision Mission" entitled the "Black Hole Imager." Diffractive/refractive systems can also be configured as high throughput flux concentrators for third generation X-ray timing studies and non-dispersive spectroscopy. They may also have an important role in very high resolution spectroscopy. Because these elements focus by transmitting X-rays at normal incidence they can be extremely lightweight compared to grazing incidence telescopes. Furthermore the figure accuracy and surface smoothness are much less critical. On the other hand diffractive/refractive optics are characterized by chromatic aberration, which can be corrected only in small wavelength bands. Focal lengths range up to 105 km making the application of diffractive/refractive X-ray optics to astronomy dependent upon the development of technology for formation flying of very widely separated spacecraft.© (2004) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.