We have performed a comprehensive study of the UV emission detected from asymptotic giant branch (AGB) stars by the Galaxy Evolution Explorer (GALEX). Of the 468 AGB stars in our sample, 316 were observed by GALEX. In the near-UV (NUV) bandpass ({lambda}_eff_~2310{AA}), 179 AGB stars were detected and 137 were not detected. Only 38 AGB stars were detected in the far-UV (FUV) bandpass ({lambda}_eff_~1528{AA}). We find that NUV emission is correlated with optical to near-infrared emission, leading to higher detection fractions among the brightest, and hence closest, AGB stars. Comparing the AGB time-variable visible phased light curves to corresponding GALEX NUV phased light curves, we find evidence that for some AGB stars the NUV emission varies in phase with the visible light curves. We also find evidence that the NUV emission and possibly the FUV emission are anticorrelated with the circumstellar envelope density. These results suggest that the origin of the GALEX-detected UV emission is an inherent characteristic of the AGB stars that can most likely be traced to a combination of photospheric and chromospheric emission. In most cases, UV detections of AGB stars are not likely to be indicative of the presence of binary companions.