The Poisson distribution is a popular distribution for modeling count data, yet it is constrained by its equi-dispersion assumption, making it less than ideal for modeling real data that often exhibit over- or under-dispersion. The COM-Poisson distribution is a two-parameter generalization of the Poisson distribution that allows for a wide range of over- and under-dispersion. It not only generalizes the Poisson distribution, but also contains the Bernoulli and geometric distributions as special cases. This distribution‟s flexibility and special properties has prompted a fast growth of methodological and applied research in various fields. This paper surveys the different COM-Poisson models that have been published thus far, and their applications in areas including marketing, transportation, and biology, among others.