The inflationary universe scenario was devised by Guth to provide a resolution to two major puzzles in the standard model of the universe, namely the horizon and flatness problems. There are two basic versions of cosmological inflation which the authors shall refer to as ‘old’ and ‘new’ inflation. In both versions, the universe undergoes a period of very rapid expansion driven by a large cosmological constant in the false vacuum. In this chapter, the authors discuss old inflation and its successes and shortcomings. They present an exposition of new inflation (slow roll inflation). The authors show that, as well as providing a solution to these cosmological problems, slow-roll inflation is capable of accounting for the size of the density perturbations in the cosmic microwave background radiation. Since all pre-existing baryon asymmetry will be diluted exponentially by the inflationary period, the baryon asymmetry the authors observe now must be generated after reheating has occurred or during reheating.