The term ”harmonic analysis” is a flexible one that has been used to denote a lot of different things. In this book I take it to mean those parts of analysis in which the action of a locally compact group plays an essential role： more specifically， the theory of unitary representations of locally compact groups， and the analysis of functions on such groups and their homogeneous spaces.
The purpose of this book is to give an exposition of the fundamen-tal ideas and theorems of that portion of harmonic analysis that can be developed with minimal assumptions on the nature of the group with which one is working. This theory was mostly developed in the period from 1927 （the date of the Peter-Weyl theorem） through the 1960s. Since that time， research in harmonic analysis has proceeded in other direc-tions， mostly on a more concrete level， so one may ask what is the excuse for a new book on the abstract theory at this time.
Well， in the first place， I submit that the material presented here is beautiful. I fell in love with it as a student， and this book is the fulfillment of a long-held promise to myself to return to it. In the second place，the abstract theory is still an indispensable foundation for the study of concrete cases; it shows what the general picture should look like and provides a number of results that are useful again and again. Moreover the intervening years have produced few if any books with the scope of the present one. One can find expositions of various bits and pieces of this subject in a lot of places， and there are a few lengthy treatises in which one can perhaps learn more about certain aspects of it than one wants to know.
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