Here's another neat use of the COM-Poisson distribution (a distribution for count data that includes as special cases Poisson, Bernoulli, and geometric distributions): a count data process! Useful for count data that are over- or under-dispersed.
Our co-authored paper Bridging the Gap: A Generalized Stochastic Process For Count Data (with Li Zhu, Kim Sellers and Darcy Morris) is now published in The American Statistician.

[The link provides free access to the first 50 readers]