Abstract: This paper proposes a new model for autoregressive conditional heteroscedasticity and kurtosis. Via a time-varying degrees of freedom parameter, the conditional variance and conditional kurtosis are permitted to evolve separately. The model uses only the standard Student's t density and consequently can be estimated simply using maximum likelihood. The method is applied to a set of four daily financial asset return series comprising US and UK stocks and bonds, and significant evidence in favour of the presence of autoregressive conditional kurtosis is observed. Various extensions to the basic model are examined, and show that conditional kurtosis appears to be positively but not significantly related to returns, and that the response of kurtosis to good and bad news is not significantly asymmetric. A multivariate model for conditional heteroscedasticity and conditional kurtosis, which can provide useful information on the co-movements between the higher moments of series, is also proposed.