In one-parameter exponential families such as the binomial and Poisson, the variance is a function of the mean. Double exponential families allow the introduction of a second parameter that controls variance independently of the mean. Double families are used as constituent distributions in generalized linear regressions, in which both means and variances are allowed to depend on observed covariates. The theory is applied to two examples—a logistic regression and a large two-way contingency table. In such cases the binomial model of variance is often untrustworthy. For example, because genuine random sampling was infeasible, the subjects may have been obtained in clumps so that the statistician should really be using smaller sample sizes. Clumped sampling is just one of many possible causes ofoverdispersion, a habitual source of concern to users of binomial and Poisson models. This article concerns a class of regression families that allow the statistician to model overdispersion while carrying out the usual regression analyses for the mean as a function of the predictors. Close connections with previous ideas concerning generalized linear models are discussed.