Abstract: The nature of the mathematical description of the evolution of open quantum systems in terms of dynamical maps has been a point of contention in the literature for over half a century now. Does the dynamical map indeed have to be completely positive? If not, does it have to be positivity preserving at least? Under what assumptions can one get complete positivity and when does it fail to be so? How does one get a physical understanding of the case where the dynamics is described not by completely positive maps? A review of these questions is presented and the explain how quantum information theory has provided several useful tools to analyse them leading to a better understanding. Some recent results regarding this question are also presented.