Gamma ray spectroscopy data acquired by Lunar Prospector are used to determine global maps of the elemental composition of the lunar surface. Maps of the abundance of major oxides, MgO, Al2O3, SiO2, CaO, TiO2, and FeO, and trace incompatible elements, K and Th, are presented along with their geochemical interpretation. Linear spectral mixing is used to model the observed gamma ray spectrum for each map pixel. The spectral shape for each elemental constituent is determined by a Monte Carlo radiation transport calculation. Linearization of the mixing model is accomplished by scaling the spectral shapes with lunar surface parameters determined by neutron spectroscopy, including the number density of neutrons slowing down within the surface and the effective atomic mass of the surface materials. The association of the highlands with the feldspathic lunar meteorites is used to calibrate the mixing model and to determine backgrounds. A linear least squares approach is used to unmix measured spectra to determine the composition of each map pixel. The present analysis uses new gamma ray production cross sections for neutron interactions, resulting in improved accuracy compared to results previously submitted to the Planetary Data System. Systematic variations in lunar composition determined by the spectral unmixing analysis are compared with the lunar soil sample and meteorite collections. Significant results include improved accuracy for the abundance of Th and K in the highlands; identification of large regions, including western Procellarum, that are not well represented by the sample collection; and the association of relatively high concentrations of Mg with KREEP-rich regions on the lunar nearside, which may have implications for the concept of an early magma ocean.