This review covers all aspects of the microwave spectrum lines seen from interstellar atoms and molecules except that nothing is said of observational methods. The discussion is concerned with general principles rather than with detailed results for individual sources. After introductory sections on the nature of interstellar matter and the types of atomic and molecular structure which produce microwave lines, the observations and their interpretation are presented. The mechanisms for line formation in the 21 cm line of atomic hydrogen are considered; observations in this line give information about the large-scale distribution of atomic hydrogen in the Galaxy which, however, appears not to be typical of the general distribution of matter, and about the structure of low-density interstellar clouds. Different theories of the nature of the high-velocity hydrogen clouds are discussed. Study of the hydrogen and helium recombination lines gives information about ionized hydrogen regions; the 'anomalous' carbon lines and some of the hydrogen results concern regions of neutral hydrogen. T h e complex array of interstellar molecules, including many organic molecules, relates mainly to higher-density clouds and to regions involved in star formation, although some molecules, such as CO, are widely distributed. Important theoretical topics concern processes of molecular formation and the pumping mechanisms involved in the non-thermal emission seen from many of the molecules. Extragalactic observations are discussed concerning the atomic hydrogen content and distribution in other galaxies, and the status of attempts to detect intergalactic neutral hydrogen in relation to cosmology and to the distances of quasars. The final sections consider the relation of the organic molecules observed to processes possibly involved in the origins of life, and attempts to estimate the prospects for communication with other civilizations in the Galaxy.