Count data commonly arise as a simple count of events in a fixed interval or the number of successes for a set of categories in a fixed number of trials. The Poisson, binomial and multinomial distributions are traditionally used to model such count data, where the appropriate choice depends on the data generating mechanism. In practice, data often exhibit over- or under-dispersion where variability observed in the data cannot be adequately captured via these standard distributions. The Conway-Maxwell (COM)-Poisson distribution supports such flexibility relative to the Poisson distribution for modeling simple count data. Shmueli et. al. (2005) present a COM-binomial distribution that permits flexibility in modeling binomial data, based on COM-Poisson conditional probabilities. We formally extend the COM-binomial distribution to the setting of more than two categories, thus defining a COM-multinomial distribution. We describe properties and illustrate the flexible characteristics of this distribution.