Most hypotheses for the origin of the Moon (rotational fission, co-accretion, and collisional ejection from the Earth, including “giant impact”) call for the formation of the Moon in a geocentric environment. However, key geochemical data for basaltic rocks from the Moon, Earth, the howardite-eucrite-diogenite (HED) meteorite parent body (probably asteroid 4-Vesta), and the shergottite-nakhlite-chassignite (SNC) meteorite parent body (likely Mars), provide no evidence that the Moon was derived from the Earth, and suggest that some objects with lunar-like compositions were produced without involvement of the Earth. The source region compositions of basalts produced in the Moon (mare basalts) were similar to those produced in the HED asteroid (eucrites) with regard to volatile-lithophile elements (Na, K, Rb, Cs, and Tl), siderophile elements (Ni, Co, Ga, Ge, Re, and Ir), and ferromagnesian elements (Mg, Fe, Cr, and V), and less similar to those in the Earth or Mars. Mare and eucrite basalts differ in their Mn abundances, Fe/Mn values, and isotopic composition, suggesting that the Moon and HED asteroid formed in different nebular locations. However, previous claims that the Moon and HED parent body differ significantly in the abundances of some elements, such as Ni, Co, Cr, and V, are not supported by the data. Instead, Cr-Mg-Fe-Ni-Co abundance systematics suggest a close similarity between the source region compositions and conditions involved in producing mare and eucrite basalts, and a significant difference from those of terrestrial basalts. The data imply that the Moon and HED asteroid experienced similar volatile-element depletion and similar fractionation of metallic and mafic phases. Among hypotheses of lunar origin, rotational fission, and small-impact collisional ejection seem less tenable than co-accretion, capture, or a variant of giant-impact collisional ejection in which the Moon inherits the composition of the impactor. Both the Moon and HED asteroid may have been derived from a class of objects that were common in the early solar system. “The most plausible model for the origin of the Moon in line with geochemical and cosmochemical constraints is an impact-induced “fission” of the proto-Earth.” — Wänke and Dreibus (1986) “Clearly, the Moon and eucrite parent body resemble each other to a high degree. Nature produced such a composition not once but at least twice. This calls into question an entire class of models that invoke ad hoc processes to explain the Moon by a unique chance event.” — Anders (1977) “The similarity between eucrites and lunar mare basalts are remarkable. Were it not for the differences in age and oxygen-isotope signature, it might be difficult to distinguish them on petrological or geochemical grounds.” — Taylor (1986)