High-precision astrometry requires accurate point-spread function modeling and accurate geometric-distortion corrections. This paper demonstrates that it is possible to achieve both requirements with data collected at the High Acuity Wide-field K-band Imager (HAWK-I); a wide-field imager installed at the Nasmyth focus of UT4/VLT ESO 8m telescope. Our final astrometric precision reaches ~3mas per coordinate for a well exposed star in a single image, with a systematic error less than 0.1mas. We constructed calibrated astro-photometric catalogs and atlases of 7 fields: the Baade's window, NGC 6656, NGC 6121, NGC 6822, NGC 6388, NGC 104, and the James Webb Space Telescope calibration field (in the LMC). We make these catalogs and images electronically available to the community. Furthermore, as a demonstration of the efficacy of our approach, combining archival material taken with the optical wide-field imager at the MPI/ESO 2.2m with HAWK-I observations, we are able to achieve an excellent separation between cluster members and field objects for NGC 6656 and NGC 6121 with a time base-line of about 8 years. Using both HST and HAWK-I data, we also study the radial distribution of the SGB populations in NGC 6656 and conclude that the radial trend is flat, within our uncertains. We also provide membership probabilities for most of the stars in NGC 6656 and NGC 6121 catalogs and estimate membership for the published variable stars in these two fields.