The basic principle of x-ray spectroscopy was discovered as early as 1914, but most of the progress in this field has taken place since 1945. The author gives the basic theory. In early experiments the specimen to be examined was made the target of the x-ray tube, or was smeared on the target, and produced x-rays known as primary radiation. In more recent work, the specimen is outside the tube and is irradiated by x-rays from the tube which cause it to fluoresce with x-rays or secondary radiation. The scope and limit of the method is discussed and a comparison is made with optical spectroscopy. The x-ray method is particularly valuable for detection of elements above atomic number 20, present in concentrations of 0.1 100%, whereas the optical method is best for trace amounts from 0.0005 5% of most elements. X-ray fluorescence spectroscopy is non-destructive to the sample as compared with the optical spark and can be faster; the spectral line structure is simpler in the x-ray region; direct-reading instruments are now becoming available. Future developments may include extension to elements of lower atomic number. Radio-isotopes may be used as the source of x-rays, so that the apparatus would be less cumbersome. One of the most important recent developments is that of the micro-probe, a beam of electrons of as little as 1 micron in diameter focussed on the specimen and the resulting characteristic x-rays detected by a bent-crystal spectrometer. Some applications relating to the work at the Research Laboratory for Archaeology and History of Art, Oxford are mentioned. One of the earliest uses was for the analysis of the Piltdown remains. The method has also been used for analysis of buried coins, where surface enrichment of one metallic constituent at the expense of another might give a method for dating. It is also more convenient for analysis of glass and ceramics than are wet methods. Examples are cited of a difference between blue glazes in early and late Ming porcelain and a difference between early Chinese and early oriental glass, both being established by this method. Very little material and very low concentrations (as little as 10^-8 g. of many elements, in solution applied to filter paper) are needed for identification by this method. -- AATA