We present the results of a new, ultra-deep, near-infrared imaging survey executed with the Hawk-I imager at the ESO VLT, of which we make all the data public. This survey, named HUGS (Hawk-I UDS and GOODS Survey), provides deep, high-quality imaging in the K and Y bands over the CANDELS UDS and GOODS-South fields. We describe here the survey strategy, the data reduction process, and the data quality. HUGS delivers the deepest and highest quality K-band images ever collected over areas of cosmological interest, and ideally complements the CANDELS data set in terms of image quality and depth. The seeing is exceptional and homogeneous, confined to the range 0.38-0.43. In the deepest region of the GOODS-S field, (which includes most of the HUDF) the K-band exposure time exceeds 80 hours of integration, yielding a 1-sigma magnitude limit of ~28.0 mag/sqarcsec. In the UDS field the survey matches the shallower depth of the CANDELS images reaching a 1-sigma limit per sq.arcsec of ~27.3mag in the K band and ~28.3mag in the Y-band, We show that the HUGS observations are well matched to the depth of the CANDELS WFC3/IR data, since the majority of even the faintest galaxies detected in the CANDELS H-band images are also detected in HUGS. We present the K-band galaxy number counts produced by combining the HUGS data from the two fields. We show that the slope of the number counts depends sensitively on the assumed distribution of galaxy sizes, with potential impact on the estimated extra-galactic background light (abridged).
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