Autonomous experimentation (AE) is an emerging paradigm that seeks to automate the entire workflow of an experiment, including---crucially---the decision-making step. Beyond mere automation and efficiency, AE aims to liberate scientists to tackle more challenging and complex problems. We describe our recent progress in the application of this concept at synchrotron x-ray scattering beamlines. We automate the measurement instrument, data analysis, and decision-making, and couple them into an autonomous loop. We exploit Gaussian process modeling to compute a surrogate model and associated uncertainty for the experimental problem, and define an objective function exploiting these. We provide example applications of AE to x-ray scattering, including imaging of samples, exploration of physical spaces through combinatorial methods, and coupling to in-situ processing platforms. These uses demonstrate how autonomous x-ray scattering can enhance efficiency, and discover new materials.