There is a great effort worldwide to develop new X‐ray microscopes both of the imaging and of the scanning type. Undulator radiation from the next generation of high brightness synchrotron radiation storage rings will be the ideal source for such microscopes. As imaging elements, Fresnel zone plates, grazing incidence mirrors and multi‐layer coated normal incidence mirrors are being developed and applied. The engineering of these elements is determined by accurate knowledge of optical constants of the materials involved and by manufacturing accuracy. The major projects are reviewed. Finally, details and first results of the Hamburg focusing mirror scanning microscope are presented. With this instrument, imaging and microprobe analysis with less than 1&mgr; resolution is possible in the energy range 20–1300 eV. Long term improvement could yield a resolution of 0.1 &mgr;m.