We report on our progress in research and development of ultrafast superconducting single-photon detectors (SSPDs) based on ultrathin NbN nanostructures. Our SSPDs were made of the 4-nm-thick NbN films with Tc ~11 K, patterned as meander-shaped, 100-nm-wide strips, and covering an area of 10×10 μm2. The detectors exploit a combined detection mechanism, where upon a single-photon absorption, a hotspot of excited electrons and redistribution of the biasing supercurrent, jointly produce a picosecond voltage transient signal across the superconducting nanostripe. The SSPDs are typically operated at 4.2 K, but their sensitivity in the infrared radiation range can be significantly improved by lowering the operating temperature from 4.2 K to 2 K. When operated at 2 K, the SSPD quantum efficiency (QE) for visible light photons reaches 30-40%, which is the saturation value limited by the optical absorption of our 4-nm-thick NbN film. With the wavelength increase of the incident photons,the QE of SSPDs decreases significantly, but even at the wavelength of 6 μm, the detector is able to count single photons and exhibits QE of about 10-2 %. The dark (false) count rate at 2 K is as low as 2x10-4 s,-1 which makes our detector essentially a background-limited sensor. The very low dark-count rate results in a noise equivalent power (NEP) below 10-18 WHz-1/2 for the mid-infrared range (6 μm). Further improvement of the SSPD performance in the mid-infrared range can be obtained by substituting NbN for another, lower-Tc materials with a narrow superconducting gap and low quasiparticles diffusivity. The use of such superconductors should shift the cutoff wavelength below 10 μm.