The absorption and fluorescence spectra of trivalent rare earth ions in crystals at low temperatures are characterized by groups of very sharp lines in the visible, near ultraviolet and infrared. It is generally agreed that these sharp line spectra are due to transitions between states of the 4fn configuration of the ion which is partially shielded from fluctuations in the surrounding crystalline field by the outer closed 5s²p⁶ shells. The average field, however, penetrates the ion and results in a crystalline Stark splitting of the free ion levels, each level of the free ion being characterized by the quantum numbers L S J. The magnitude of the Stark splittings of the free ion levels, and the resultant number of components into which the levels are split depend on the nature of the crystal field and the L S J state of the level. By studying the behaviour of particular L S J states of the free ions on immersion in a crystalline field a considerable amount of information on the nature of the crystal field could be gained. Unfortunately, the L S J states of the free rare earth ions are unknown apart from the trivial case of 4f¹ (Ce³⁺). Thus, before the rare earth ions can be profitably used as probes, an attempt must be made to determine the L S J designations of the individual groups of lines observed in the solid state. The research reported here represents a successful attempt to assign L S J values to the observed line groups of many of the rare earth salts. In chapter one the broad field of the solid state spectra of the rare earth ions and their interpretation will be sketched while the succeeding chapters will outline the calculation of the energy levels of 4fn configurations and the application of these calculations to the interpretation and assignment of L S J values to known line groups of their spectra.