The aim of this paper is to provide a fundamental theoretical elaboration of infinitely divisible distributions. For this purpose, the concept of an infinitely divisible distribution and characteristic function is first defined, including basic properties and examples. Mathematical tools for their construction, which yield significant results about their structure, are presented and discussed. An alternative approach to defining infinite divisibility using generating functions is presented. The importance of this class for modeling stochastic processes with stationary independent increments is illustrated using the example of the Poisson process. The canonical representations of infinitely divisible characteristic functions are studied, with particular emphasis on the Levy-Khintchine theorem. The relationship between the convergence of infinitely divisible distributions and the associated Levy-Khintchine pairs is analyzed. In the final part of the paper, stable and geometrically infinitely divisible distributions are presented.