During its all-sky survey, the Extreme Ultraviolet Explorer (EUVE) satellite observed the Moon several times at first and last quarters, and once immediately following the Dec. 10, 1992 lunar eclipse. We present here a portion of this data, in the form of extreme-ultraviolet (EUV) images of the Moon and derived geometric albedos. Extreme ultraviolet observations of the Moon are of considerable interest, since it has been speculated that lunar EUV emissions may be dominated by L- and M-shell X-ray florescence and may thus provide a useful diagnostic of surface elemental abundances. From the EUVE photometer data we obtain average geometric albedos of 0.15% (¿0.03%), 3.1% (¿0.3%), and 3.5% (¿0.3%), over wavelength intervals of 150--240 ¿, 400--580 ¿, and 550--650 ¿, respectively. The standard deviations listed result from photon counting statistics only-uncertainties arising from the assumed solar EUV flux and the correction to zero phase angle are not included. An upper limit geometric albedo of 0.13% is obtained for the wavelength interval 75--180 ¿. Also, using previously published ROSAT data, we estimate a lunar geometric albedo of 0.014% (¿0.002%) over the wavelength interval 50--80 ¿. These EUV albedos (and previously published far-ultraviolet albedos) are well fit by the scaled reflectivities of SiO2 and Al2O3. Over the wavelength ranges of the EUVE photometers, the observed brightness of the Moon seems to be largely consistent with reflected sunlight rather than x-ray fluorescence. Since the L- and M-shell fluorescence signal is expected to be carried by only small number of emission lines, however, it will require EUV observations of higher spectral resolution (≲5 ¿) to determine their exact contribution, if any, to the lunar EUV spectrum.