This paper reviews the first experiments on Mössbauer filtration of synchrotron radiation (that is, experiments on separating, out of the continuous energy spectrum of the synchrotron radiation, a beam of extremely highly monochromatized radiation with a relative linewidth of the order of 10$^{-12}$ at an energy of 14.4 keV) by coherent nuclear resonance scattering of the synchrotron radiation by the nuclei of a M\"ossbauer isotope. The corresponding theoretical investigations as well as possible variations of Mössbauer filtration of synchrotron radiation are discussed. The combination of the advantages of synchrotron radiation and Mössbauer spectroscopy, which is characterized by a uniquely high energy resolution, opens up new prospects for the use of synchrotron radiation in basic and applied investigations. This outlook refers first of all to the nuclear physics of solids and in particular to the topic of nuclear excitons in a crystal, to studies of the magnetic and electric field structure in crystals, metrological applications of synchrotron radiation, etc. The possibility of implementing new M\"ossbauer experiments with the use of synchrotron radiation is discusssed.