We develop an extension of the classical Zellner's g-prior to generalized linear models. Any continuous proper hyperprior f(g) can be used, giving rise to a large class of hyper-g priors. Connections with the literature are described in detail. A fast and accurate integrated Laplace approximation of the marginal likelihood makes inference in large model spaces feasible. For posterior parameter estimation we propose an effcient and tuning-free Metropolis-Hastings sampler. The methodology is illustrated with variable selection and automatic covariate transformation in the Pima Indians diabetes data set.