The object of this monograph Is to give an exposition of the real-variable theory of Hardy spaces (H^p spaces). This theory has attracted considerable attention in recent years because it led to a better understanding in IR^n of such related topics as singular integrals, multiplier operators, maximal functions, and real-variable methods generally. Because of its fruitful development it seems to us that a systematic exposition of some of the main parts of this theory is now desirable. There are, however, good reasons why in addition the theory should be recast in the more general setting where the underlying IR^n is replaced by a homogeneous group. The justification for this wider scope, both in terms of the structure of the theory and its applications, will be described in more detail below.

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