Studies are reviewed that employ the effect of an anomalously strong shift of the fundamental $K$-series x-ray lines associated with a change in the number of 4$f$ electrons in a given atom for investigating the phenomenon of variable valency in compounds of the rare-earth elements in general, and the so-called isomorphous phase transitions in these compounds in particular. The mechanism of the shift effect and its theory are described. The available data on the shifts of the $K$-lines of the different compounds of all the rare-earth elements (except promethium) are discussed. A summary table gives the valencies (including the anomalous ones), the occurrence of which has been proved by this method. Application of the $K$-shift method for studying isomorphous (or isostructural) transitions allows one to establish unambiguously that the mechanism of these transitions involves a change in the number of 4$f$ electrons, and to measure the magnitude of these changes. The review concludes with a brief description of the principles and design of the required experimental apparatus-crystal diffraction spectrometers.