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•Previous work has demonstrated that individual task-based brain activity can be predicted from resting-state functional connectivity.•We build on recent deep learning methods to create a surface-based fully-convolutional neural network model that works with a representation of the brain’s cortical sheet.•The proposed model, BrainSurfCNN, can achieve state of the art predictive accuracy on independent test data from the Human Connectome Project.•BrainSurfCNN yields individual-level predicted maps that are on par with the target-repeat reliability of the measured contrast maps.•We further demonstrate that BrainSurfCNN can generalize well to novel domains with limited training data. Task-based and resting-state represent the two most common experimental paradigms of functional neuroimaging. While resting-state offers a flexible and scalable approach for characterizing brain function, task-based techniques provide superior localization. In this paper, we build on recent deep learning methods to create a model that predicts task-based contrast maps from resting-state fMRI scans. Specifically, we propose BrainSurfCNN, a surface-based fully-convolutional neural network model that works with a representation of the brain’s cortical sheet. BrainSurfCNN achieves exceptional predictive accuracy on independent test data from the Human Connectome Project, which is on par with the repeat reliability of the measured subject-level contrast maps. Conversely, our analyses reveal that a previously published benchmark is no better than group-average contrast maps. Finally, we demonstrate that BrainSurfCNN can generalize remarkably well to novel domains with limited training data.