In this book I try to give a readable introduction to the modern theory of the Fourier transform and to show some interesting applications of that theory in higher analysis. The book is directed to students having only a moderate preparation in real and complex analysis. More exactly, I suppose the reader to be familiar with the elements of real variables and Lebesgue integration and to have some knowledge of analytic functions. Further along in the book both Hilbert spaces and L^p-spaces play a role, but the reader is presumed to know only a little about either topic, much less, in fact, than appears in any standard modern real variable textbook.