The Poisson process is a very useful tool for modeling random recurrent events. In general, the Poisson process is used to model the arrival process of recurrent events. Especially in reliability area, the Poisson process is usefully applied to model the minimal repair process of repairable systems. Although the Poisson process is the most frequently applied point process model, it also has its own limitations. Specifically, the Poisson process possesses the independent increments property. Furthermore, at any point of time, the variance and expectation of the number of event occurrences are the same. These can be critical restrictions in the application of the Poisson process to the practical situations where such restrictive conditions do not hold. Recently, there have been approaches to extend the Poisson process to more generalized models. In this paper, some generalized models of the Poisson process and their reliability applications are introduced and reviewed.