Structural curvatures are widely used seismic attributes that help interpreters to understand both structural and stratigraphic features. Traditional structural curvature extractions are mainly calculated from dip estimations through lateral scanning of seismic events, which is not only a very time-costing approach but also influenced by parameter settings, seismic frequency, and data quality. In this article, we propose a deep learning-based volumetric curvature extraction approach that directly derives structural curvature volumes from the seismic response. To realize the above approach, we develop a suite of sample generation and augmentation methods to synthesize seismic samples with accurate curvature labels. Then, a multitask end-to-end convolutional neural network architecture and a geometric loss function are proposed to establish the volume mapping model from complex seismic responses to the most positive and negative curvature volumes. The performance of the proposed curvature extraction approach is evaluated on both the synthetic data and the Netherlands F3 field seismic data. Extensive experiments demonstrate that curvature volumes extracted with the proposed approach are not only more accurate and less influenced by the noises of poststack seismic data but also more friendly for structure interpretation. Therefore, we believe that our proposed deep learning curvature extraction approach can be a useful tool for further seismic structure interpretation practices.