Recent portfolio-choice, asset-pricing, value-at-risk, and option-valuation models highlight the importance of modeling the asymmetry and tail-fatness of returns. These characteristics are captured by the skewness and the kurtosis. We characterize the maximal range of skewness and kurtosis for which a density exists and show that the generalized Student-t distribution spans a large domain in the maximal set. We use this distribution to model innovations of a GARCH type model, where parameters are conditional. After demonstrating that an autoregressive specification of the parameters may yield spurious results, we estimate and test restrictions of the model, for a set of daily stock-index and foreign-exchange returns. The estimation is implemented as a constrained optimization via a sequential quadratic programming algorithm. Adequacy tests demonstrate the importance of a time-varying distribution for the innovations. In almost all series, we find time dependency of the asymmetry parameter, whereas the degree-of-freedom parameter is generally found to be constant over time. We also provide evidence that skewness is strongly persistent, but kurtosis is much less so. A simulation validates our estimations and we conjecture that normality holds for the estimates. In a cross-section setting, we also document covariability of moments beyond volatility, suggesting that extreme realizations tend to occur simultaneously on different markets.