This paper is an overview of the research activities carried out in the past five years at the Institute for Physics of Microstructures RAS and "X-ray" Company towards the manufacture of multilayer mirror systems capable of forming X-ray beams in the subnanometer range of wavelengths. The systems fabrication technology is presented, including techniques for producing supersmooth surfaces of specified shape, methods of graded multilayer structure deposition on such surfaces, and the principles of designing optimal mirror parameters. The characteristics of a quadrelliptical reflector -a novel high light-gathering power four-corner focusing system -are reported.