ABSTRACT The Large Binocular Telescope Interferometer (LBTI) is a versatile instrument designed for high angular resolution and high-contrast infrared imaging (1.5-13 m). In this paper, we focus on the mid-infrared (8-13 m) nulling mode and present its theory of operation, data reduction, and on-sky performance as of the end of the commissioning phase in 2015 March. With an interferometric baseline of 14.4 m, the LBTI nuller is specifically tuned to resolve the habitable zone of nearby main-sequence stars, where warm exozodiacal dust emission peaks. Measuring the exozodi luminosity function of nearby main-sequence stars is a key milestone to prepare for future exo-Earth direct imaging instruments. Thanks to recent progress in wavefront control and phase stabilization, as well as in data reduction techniques, the LBTI demonstrated in 2015 February a calibrated null accuracy of 0.05% over a 3 hr long observing sequence on the bright nearby A3V star β Leo. This is equivalent to an exozodiacal disk density of 15-30 zodi for a Sun-like star located at 10 pc, depending on the adopted disk model. This result sets a new record for high-contrast mid-infrared interferometric imaging and opens a new window on the study of planetary systems.