This book is issued in the Springer Universitext series with a moderate size, 172 p. The subject is purely not new and some classical books have been dealt with, see the books of S. Bochner, E. C. Titchmarsh, I. N. Sneddon. The text is divided into three parts: (1) Fourier transforms on \(L_ 1({\mathbb{R}})\); (2) Fourier transforms on \(L_ 2({\mathbb{R}})\); (3) Fourier Stieltjes transforms (one variable). Bibliographical notes and References are also included. Part (1) contains the problems pertaining to basic properties, localization principle, Mellin inversion formula, series summability and some tauberian theorems. Part (2) contains the discussion of Planchérel theory, translation closure properties, Paley- Wiener theorem on bounded spectrum functions. In both parts the possible extensions for several variables are presented. Part (3) gives the exposition of the theory of characteristic and distribution functions and Bochner and Riesz theorems on positive definite functions. The text is well organized, splendidly written and constitutes a beautiful introduction to Fourier transform theory. This stems, undoubtly, from the fact that the author is a well known name in this area. Finally, we think the book will soon become an excellent introductory text for a large class of readers.