This tract gives a clear exposition of the elementary theory of Fourier transforms, so arranged as to give easy access to the recently developed abstract theory of Fourier transforms on a locally compact group. (This latter subject has important applications to the general treatment of unitary representations of the rotation group, the Lorentz group and other classical groups that is of value in quantum field theory and other branches of mathematical physics.) A knowledge of Lebesgue integration and, in one chapter, of Riemann-Stieltjes integration is assumed; the results needed are all stated in the introductory chapter.