It is not the object of the author to present comprehensive covยญ erage of any particular integral transformation or of any particular development of generalized functions, for there are books available in which this is done. Rather, this consists more of an introductory survey in which various ideas are explored. The Laplace transformaยญ tion is taken as the model type of an integral transformation and a number of its properties are developed; later, the Fourier transforยญ mation is introduced. The operational calculus of Mikusinski is preยญ sented as a method of introducing generalized functions associated with the Laplace transformation. The construction is analogous to the construction of the rational numbers from the integers. Further on, generalized functions associated with the problem of extension of the Fourier transformation are introduced. This construction is analยญ ogous to the construction of the reals from the rationals by means of Cauchy sequences. A chapter with sections on a variety of transยญ formations is adjoined. Necessary levels of sophistication start low in the first chapter, but they grow considerably in some sections of later chapters. Background needs are stated at the beginnings of each chapter. Many theorems are given without proofs, which seems approยญ priate for the goals in mind. A selection of references is included. Without showing many of the details of rigor it is hoped that a strong indication is given that a firm mathematical foundation does actuยญ ally exist for such entities as the "Dirac delta-function"