The high-precision gamma-ray spectrometer (GRS) on the lunar polar orbiter SELENE is designed to measure 200 keV-12 MeV gamma rays in order to determine elemental compositions of the lunar surface. The GRS consists of a large germanium (Ge) crystal as a main detector and a massive bismuth germanate crystal and a plastic scintillator as anticoincidence detectors. The Ge detector is cooled by a Stirling cryocooler with its compressor attached to a passive radiator facing the cold space. The cooling system maintains the Ge detector below 90 K during the observation. The flight model of the GRS has achieved an energy resolution of 3.0 keV (FWHM) at 1333 keV. Energy spectra obtained by the GRS will show sharp gamma-ray lines whose energies identify the elements and whose intensities determine the concentrations of the elements, permitting global mapping of the elemental abundances in the sub-surface of the Moon. The elemental maps obtained by the GRS with such high-energy resolution enable us to study lunar geoscience problems. Key words: Moon, gamma-ray spectroscopy, chemical composition, lunar formation and evolution, SELENE (KAGUYA), GRS.