Cherenkov radiation in the soft x-ray region is generated in narrowband regions at inner-shell absorption edges. Mainly low-Z elements are suitable Cherenkov sources, which emit in a photon energy range from 30 eV to 1 keV and require moderate electron energies up to 25 MeV. Generally, in the soft x-ray region materials are highly absorbing and therefore the Cherenkov radiation theory is discussed for absorbing media. A detailed description includes transition radiation that is generated at the interface when the relativistic electron exits the material. We show that the transition radiation yield equation, when it is adopted for an absorbing medium, includes Cherenkov radiation. Based on this approach it is shown that the spectral intensity of Cherenkov radiation in the soft x-ray region is large compared to transition radiation for moderate electron energies. First measurements of soft x-ray Cherenkov radiation in the water-window spectral region, generated in titanium and vanadium foils, are discussed in detail. The measured spectral and angular distribution of the radiation, and the measured total yield (≈ 10 -4 photon per electron) are in agreement with theoretical predictions based on the refractive index data. We show that the brightness that can be achieved using a small electron accelerator is sufficient for practical x-ray microscopy in the water window.