Motivated by the need for reliable specification of the Sun's electromagnetic radiation in the extreme ultraviolet (EUV) spectrum, we have developed a new model of solar EUV irradiance variability at wavelengths from 50 to 1200 ¿. Solar images are used to quantify changes in the sources of EUV irradiance during the solar cycle. Optically thin EUV emission line fluxes are estimated from differential emission measures (DEMs) that characterize the properties of the solar atmosphere in the source regions, while fluxes for optically thick lines are modeled directly by specifying the source region contrasts. We compare the new model, NRLEUV, with three different empirical models of solar EUV irradiance since 1975. For solar cycles 21 and 22, NRLEUV predicts overall lower EUV irradiances and smaller solar cycle variability than the empirical models. The average total EUV energy at wavelengths from 50 to 1050 ¿ is 2.9 mW m-2, smaller than the HFG, EUVAC, and SOLAR2000 models for which average energies are 3.7, 4.3, and 5.6 mW m-2, respectively. These differences have distinct wavelength dependencies. The solar cycle variation in total EUV energy is 1.9 for NRLEUV compared with 2.7, 2.9, and 2.3 for HFG, EUVAC, and SOLAR2000. Here, too, the differences are wavelength dependent. We compare both the NRLEUV and the empirically modeled EUV irradiances with selected wavelength bands and emission lines measured during 4 years in cycle 21 by Atmospheric Explorer-E (AE-E) and two broad bands at 170--200 and 260--340 ¿ measured in cycle 23 by the Solar X-Ray Photometer (SXP) on the Student Nitric Oxide Experiment (SNOE) and the Solar EUV Monitor (SEM) on the Solar and Heliospheric Observatory (SOHO), respectively. The NRLEUV model reproduces the variations observed during solar rotation better than, or as well as, the empirical models. Comparisons of solar cycle variations are more ambiguous because undetected instrumental drifts can cause spurious trends in the observations over these longer timescales. Drifts in the AE-E instruments may explain why the HFG and EUVAC models, which are based on parameterizations of these data, have larger solar cycle variations than NRLEUV. We assess the implications for space weather of the significant differences among the modeled EUV irradiances by using the Atmospheric Ultraviolet Radiance Integrated Code (AURIC) to quantify corresponding differences in upper atmosphere energy deposition and photoionization rates.