The Bernoulli and Poisson processes are two popular discrete count processes; however, both rely on strict assumptions. We instead propose a generalized homogenous count process (which we name the ConwayMaxwell- Poisson or COM-Poisson process) that not only includes the Bernoulli and Poisson processes as special cases, but also serves as a flexible mechanism to describe count processes that approximate data with over-or under-dispersion. We introduce the process and an associated generalized waiting time distribution with several real-data applications to illustrate its flexibility for a variety of data structures. We consider model estimation under different scenarios of data availability, and assess performance through simulated and real datasets. This new generalized process will enable analysts to better model count processes where data dispersion exists in a more accommodating and flexible manner.