Since 1945 there has been a vigorous development in infrared instrumentation. Commercial spectrometers are now available that can resolve the rotational structure of vibrational bands of.lighter molecules, and many research instruments are available that can resolve the rotational structure in the spectra. of moderately heavy molecules. Much of the basic theory for the interpretation of these spectra was developed during the early years of quantum mechanics. The.available knowledge up to 1945 is well covered in the excellent book by Herzberg. - In the past fifteen years fIluch of this theory has been correlated and extended. Unfortunately, most of this work was scattered throughout the literature. It is our purpose to present this material in an orderly manneF, to relate the diverse notations, and to summarize some of the accumulated experience in applying the theory to the analysis of observed spectra. The treatment in the first five. chapters is intended to give a consistent development of the basic theory. The aim has been to present the theory with sufficient completeness to enable the reader to understand its derivation and to be prepared to extend and adapt the methods involved to new problems. No attempt has been made to include all the esoteric extensions of he theory that have been made in the past few years, since these treatments are so detailed that the notation becomes ponderous and confusing to the beginner.