We report our studies on the spectral sensitivity of superconducting NbN thin-film single-photon detectors (SPD's) capable of GHz counting rates of visible and near-infrared photons. In particular, it has been shown that a NbN SPD is sensitive to 1.55-µm wavelength radiation and can be used for quantum communication. Our SPD's exhibit experimentally measured intrinsic quantum efficiencies from 20% at 800 nm up to 1% at 1.55-µm wavelength. The devices demonstrate picosecond response time (<100 ps, limited by our readout system) and negligibly low dark counts. Spectral dependencies of photon counting of continuous-wave, 0.4-µm to 3.5-µm radiation, and 0.63-µm, 1.33-µm, and 1.55-µm laser-pulsed radiations are presented for the single-stripe-type and meander-type devices.