This is the second edition of a bestseller by the first author, which was published in 1995. The book is aimed as a senior undergraduate or graduate text on integral transforms and has already been widely used for such a purpose. The level of explanation is accessible for a large audience of interested readers; the careful treatment of the basic methods (avoiding excessive abstractness in their description) is accompanied by a great number of selected examples developing the reader's analytical skills, ranging from applications of Laplace transforms in the theory of linear differential equations through to fluid mechanics. The subject treated in the book belongs to the very core of classical analysis and at the same time it is a vivid field with many recent important developments. This is also reflected in the contents of the second edition where new chapters on fractional calculus, Radon transforms and wavelets have been added. A few keywords from the 19 chapters of the book are: Fourier transforms, Laplace transforms, fractional calculus, Hankel, Mellin, Hilbert and Stieltjes transforms, Z transforms, Jacobi and Gegenbauer, Laguere, Hermite transforms, Radon transforms, wavelets, and some special functions. The book is accompanied by tables of transforms of important functions and by hints to selected exercises. Among other existing treatises on integral transforms this book surely deserves attention, especially from the reader who is a novice in the subject looking for an accessible introduction to this vast area of classical and modern analysis.