Superconducting nanowire single-photon detectors (SNSPDs) perform single-photon counting with exceptional sensitivity and time resolution at near-infrared wavelengths. State-of-the-art SNSPDs, based on 100 nm-wide, 4 to 5 nm thick NbN nanowires, are vulnerable to constrictions, which significantly limit their yield. Also, their sensitivity becomes negligible beyond 2 mu m wavelength, which makes them unsuitable for mid-infrared applications. SNSPDs based on few-tens-of-nanometer-wide nanowires are expected to efficiently detect mid-infrared photons and to operate at low bias currents, so constrictions may have less impact on their performance. Prior to this work, SNSPDs based on nanowires narrower than 50-nm had not been demonstrated because: (1) the SNSPD signal is roughly proportional to the nanowire width, so narrow nanowires have poor signal-to-noise ratio; and (2) fabrication at these length scales is extremely challenging. In this letter we report how we addressed these issues and demonstrated single-photon detection (20% detection efficiency at 1550 nm wavelength) with 30- and 20-nm-wide-nanowire detectors.