The Poisson regression (PR) model is inappropriate for modeling over- or under-dispersed (or inflated) data. Several generalizations of PR model have been proposed for modeling such data. In this paper, a rich family of generalized Poisson regression (GPR) models is reviewed in detail. The family has a wide range of applications in various disciplines including agriculture, econometrics, patent applications, species abundance, medicine, and use of recreational facilities. For illustrating the usefulness of the family, several applications with different situations are given. For example, hospital discharge counts are modeled using GPR and other generalized models, in which the applied models show that household size, education, and income are positively related to diagnosis-related groups (DRGs) hospital discharges. One of the advantages of using the family is that it lets data determine which model is appropriate for a given situation. It is expected that the results discussed in the paper would enhance our understanding of various forms of count data originating from primary health care facilities and medical domains. © 2005 IMACS. Published by Elsevier B.V. All rights reserved.