We present a new dataset and benchmark with the goal of advancing research in the intersection of brain activities and eye movements. Our dataset, EEGEyeNet, consists of simultaneous Electroencephalography (EEG) and Eye-tracking (ET) recordings from 356 different subjects collected from three different experimental paradigms. Using this dataset, we also propose a benchmark to evaluate gaze prediction from EEG measurements. The benchmark consists of three tasks with an increasing level of difficulty: left-right, angle-amplitude and absolute position. We run extensive experiments on this benchmark in order to provide solid baselines, both based on classical machine learning models and on large neural networks. We release our complete code and data and provide a simple and easy-to-use interface to evaluate new methods.