We present the unTimely Catalog, a deep time-domain catalog of detections based on Wide-field Infrared Survey Explorer (WISE) and NEOWISE observations spanning the 2010 through 2020 time period. Detections are extracted from “time-resolved unWISE coadds,” which stack together each biannual sky pass of WISE imaging to create a set of ∼16 all-sky maps (per band), each much deeper and cleaner than individual WISE exposures. unTimely incorporates the W1 (3.4 μm) and W2 (4.6 μm) channels, meaning that our data set effectively consists of ∼32 full-sky unWISE catalogs. We run the crowdsource crowded-field point-source photometry pipeline (Schlafly et al. 2018) on each epochal coadd independently, with low detection thresholds: S/N = 4.0 (2.5) in W1 (W2). In total, we tabulate and publicly release 23.5 billion (19.9 billion) detections at W1 (W2). unTimely is ∼1.3 mag deeper than the WISE/NEOWISE Single Exposure Source Tables near the ecliptic, with further enhanced depth toward higher ecliptic latitudes. The unTimely Catalog is primarily designed to enable novel searches for faint, fast-moving objects, such as Y dwarfs and/or late-type (T/Y) subdwarfs in the Milky Way’s thick disk or halo. unTimely will also facilitate other time-domain science applications, such as all-sky studies of quasar variability at mid-infrared wavelengths over a decade-long time baseline.