This paper generalizes the ACD models of Engle and Russell (1998) using the so-called q-Weibull distribution as the conditional distribution. The new specification allows the hazard function to be non-monotonic. We document that the q-Weibull distribution recently suggested in physics as a generalization of the Weibull distribution is closely related to the much older Burr Type XII distribution in statistics. The nested, more heavy-tailed, q-exponential distribution is also being introduced. Data from the New York and the Helsinki Stock Exchange with different market microstructures and data types show that the q-Weibull specification outperforms the standard specifications used in econometrics and performs as well as the Burr specification of Grammig and Maurer (2000). The more parsimonious q-exponential specification typically provides a reasonable fit, improving the fit over the most commonly applied Weibull specification. We also find that the price threshold used affects the shape of the hazard function and thus the relative success of the models and must be taken into account when modeling price durations.