While Poisson regression is a popular tool for modeling count data, it is limited by its associated model assumptions. One assumption is that the response variable follows a Poisson distribution. However, over- or under-dispersion are common in practice and are not accommodated by Poisson regression. In addition, the dispersion is assumed fixed across observations, whereas in practice dispersion may vary across groups or according to some other factor. Recently, Sellers and Shmueli (2008) introduced the Conway-Maxwell-Poisson (CMP) regression, based on the CMP distribution. CMP regression generalizes both Poisson and logistic regression models and allows for over- or under-dispersed count data. The model structure introduced, however, assumes a fixed dispersion level across all observations. In this paper, we extend the CMP regression model to account for observation-level dispersion. We discuss model estimation, inference, diagnostics, and interpretation, and present a variable selection technique. We then compare our model to several alternatives and illustrate its advantages and usefulness using datasets with varying types and levels of dispersion.