We present a sample of 1317 Infrared-faint radio sources (IFRSs) that, for the first time, are reliably detected in the infrared, generated by cross-correlating the Wide-field Infrared Survey Explorer (WISE) all-sky survey with major radio surveys. Our IFRSs are brighter in both radio and infrared than the first-generation IFRSs that were undetected in the infrared by the Spitzer Space Telescope. We present the first spectroscopic redshifts of IFRSs, and find that all but one of the IFRSs with spectroscopy have z>2. We also report the first X-ray counterparts of IFRSs, and present an analysis of radio spectra and polarization, and show that they include gigahertz peaked-spectrum, compact steep-spectrum and ultra-steep-spectrum sources. These results, together with their WISE infrared colours and radio morphologies, imply that our sample of IFRSs represents a population of radio-loud active galactic nuclei at z>2. We conclude that our sample consists of lower redshift counterparts of the extreme first-generation IFRSs, suggesting that the fainter IFRSs are at even higher redshift.