This paper explores the relation between non-exponential waiting times between events and the distribution of the number of events in a fixed time interval. It is shown that within this framework the frequently observed phenomenon of overdispersion, i.e. a variance that exceeds the mean, is caused by a decreasing hazard function of the waiting times, while an increasing hazard function leads to underdispersion. Using the assumption of i.i.d. gamma distributed waiting times, a new count data model is derived. Its use is illustrated in two applications: the number of births, and the number of doctor consultations.