A radiation mechanism — Vavilov—Cherenkov radiation (VCR) in the soft X-ray region — is considered. A region of anomalous dispersion exists near the photoabsorption edges of some materials where the real part of the permittivity exceeds unity. It is in this narrow frequency range that the VCR mechanism is realized. Analytical expressions for spectral and angular distributions of VCR from a target, an inclined plate, are derived. The VCR formation length is estimated taking into account absorption in the target (radiator) material. An experiment is discussed in which VCR in the 100-eV photon energy range was examined using a 5.7-MeV electron beam from the Tomsk Polytechnic University microtron. Experimental data and theory are shown to be in satisfactory agreement.