In a regression setting, we define an I-prior as a Gaussian process prior on the regression function with covariance kernel equal to its Fisher information. We present some methodology and computational work on estimating regression functions by working in the appropriate reproducing kernel Hilbert space of functions and assuming an I-prior on the function of interest. In a regression model with normally distributed errors, estimation is simple—maximum likelihood and the EM algorithm is employed. In the classification models (categorical response models), estimation is performed using variational inference. I-prior models perform comparatively well, and often better, to similar leading state-of-the-art models for use in prediction and inference. Applications are plentiful, including smoothing models, modelling multilevel data, longitudinal data, functional covariates, multi-class classification, and even spatiotemporal modelling.