We present Galaxy Evolution Explorer (GALEX) data for 44 Galactic globular clusters (GCs) obtained during three GALEX observing cycles between 2004 and 2008. This is the largest homogeneous data set on the UV photometric properties of Galactic GCs ever collected. The sample selection and photometric analysis are discussed, and color-magnitude diagrams (CMDs) are presented. The blue and intermediate-blue horizontal branch is the dominant feature of the UV CMDs of old Galactic GCs. Our sample is large enough to display the remarkable variety of horizontal branch shapes found in old stellar populations. Other stellar types that are obviously detected are blue stragglers and post-core-He burning stars. The main features of UV CMDs of Galactic GCs are briefly discussed. We establish the locus of post-core-He burning stars in the UV CMD and present a catalog of candidate asymptotic giant branch (AGB), AGB-manque, post early-AGB, and post-AGB stars within our cluster sample.