The Heisenberg group has been implicitly present in much of mathematics and physics for a long time. Thangavelu’s work is developed within the framework of harmonic analysis (Fourier transforms, convolution algebra, and related ideas) and gives a survey of the field. Additionally the author discusses in detail the representation theory of the group and its relationship to the theory of classical special functions. Among the topics covered are the Plancherel and Paley-Wiener theorems, spectral theory of the sublaplacian, Wiener-Tauberian theorems, Bocher-Riesz means and multipliers for the Fourier transform. The exposition is systematic, and leads to several problems and conjectures for further consideration. Any reader who is interested in pursuing research on the Heisenberg group should find this text of use.