The purpose of this overview paper is to present an approach to Mellin transform theory that is fully independent of Laplace or Fourier transform results, in a unified systematic form, one that contains the transform properties and results under natural, minimal assumptions upon the functions in question. Cornerstones are two definitions of the Mellin transform, a local and a global transform, the Mellin inversion theory, established by approximation theoretical methods connected with the Mellin convolution singular integral of Gauss-Weierstrass-type, and especially the Mellin operators of differentiation and anti-differentiation. (orig.)Available from TIB Hannover: RN 2414(468) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman