The current status of the problem of obtaining high-purity silicon isotopes 28Si, 29Si, and 30Si is analyzed. The scheme of obtaining monoisotopic silicon includes the stages of isotope separation in the form SiF4, synthesis and deep purification of isotopically enriched silane, obtaining polycrystalline silicon-28,-29, and-30, and growing monocrystals. The basic problems and methods of their solution in the synthesis and deep purification of silane and obtaining poly-and monocrystals of isotopically enriched silicon are discussed. Data characterizing the achieved level of chemical and isotopic purity of high-purity monocrystals of silicon-28 with a main isotope content of more than 99.99% and silicon-29 and silicon-30 with isotopic purity higher than 99% are presented. In monocrystalline 28Si, the boron content was 4.5 x 1013, the phosphorus content was 5 x 1011, the carbon and oxygen contents were < 1 x 1016 at/cm3, and the specific resistance was 800 Omega cm. The results of investigation of heat capacity, heat conduction, photoluminescence, and electron paramagnetic resonance spectra for monoisotopic silicon-28 are presented. The heat conduction of monoisotopic silicon is increased considerably owing to the reduced photon scattering on isotopic inhomogeneities. In the region of 20-30 K, the heat conduction of silicon-28 with an isotopic purity of 99.98% is higher by a factor of 8 than the heat conduction of natural silicon. Investigations of photoluminescence spectra in the magnetic field in the low-temperature region demonstrated the capability of optical detection of nuclear spin states of a phosphorus admixture in high-purity silicon-28. p ]Topical questions for further investigations and possible fields of practical application of high-purity isotopically enriched silicon are discussed.