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Textures and edges contribute different information to image recognition. Edges and boundaries encode shape information, while textures manifest the appearance of regions. Despite the success of Convolutional Neural Networks (CNNs) in computer vision and medical image analysis applications, predominantly only texture abstractions are learned, which often leads to imprecise boundary delineations. In medical imaging, expert manual segmentation often relies on organ boundaries; for example, to manually segment a liver, a medical practitioner usually identifies edges first and subsequently fills in the segmentation mask. Motivated by these observations, we propose a plug-and-play module, dubbed Edge-Gated CNNs (EG-CNNs), that can be used with existing encoder-decoder architectures to process both edge and texture information. The EG-CNN learns to emphasize the edges in the encoder, to predict crisp boundaries by an auxiliary edge supervision, and to fuse its output with the original CNN output. We evaluate the effectiveness of the EG-CNN with various mainstream CNNs on two publicly available datasets, BraTS 19 and KiTS 19 for brain tumor and kidney semantic segmentation. We demonstrate how the addition of EG-CNN consistently improves segmentation accuracy and generalization performance.