The authors describe a systematic historical survey of the well-known Mellin transform and present a new approach to the Mellin transform that is fully independent of Laplace or Fourier transform theory in a unified form containing basic properties and major results under natural, minimal hypotheses upon the functions involved therein. The basis of the approach are two definitions of the transform, a local and global Mellin transform, the Mellin translation and convolution structure. In particular approximation-theoretical methods connected with the Mellin convolution singular integral enable one to establish the Mellin inversion theory. Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame. These operators are different from those considered so far and more general and are of particular importance in solving differential and integral equations. As applications, the wave equation on \(\mathbb{R}_+\times \mathbb{R}_+\) and the heat equation in a semi-infinite rod are solved in detail.