The present paper outlines the requirements for the performance of x-ray mirrors at the European Synchrotron Radiation Facility (ESRF), to be built in Grenoble, France. It is shown that present-day surface preparation techniques are about adequate to achieve conservation of the source emittance, although some improvements are needed in special cases. It is much harder to conserve brilliance, where thermal deformation is the major obstacle. Here substantial research and development efforts are absolutely indispensable. Two possible ways are indicated to solve the heat problem: cryogenic cooling of silicon-based mirrors and adaptive optical systems. In the first case thermal deformations are drastically reduced, and in the second they can be compensated by mechanical forces. Our results are based on theoretical considerations of scattering by nonideal surfaces and on a thermomechanical analysis, which are also given. For layered synthetic microstructures the technological problems appear to be still more difficult. Because the critical photon energy of the ESRF 6 GeV storage ring and of most of its insertion devices is between 10 and 20 keV or even higher, the discussion is limited to hard x-ray optics.