Chest X-ray (CXR) is one of the most commonly prescribed medical imaging procedures, often with over 2-10x more scans than other imaging modalities such as MRI, CT scan, and PET scans. These voluminous CXR scans place significant workloads on radiologists and medical practitioners. Organ segmentation is a crucial step to obtain effective computer-aided detection on CXR. In this work, we propose Structure Correcting Adversarial Network (SCAN) to segment lung fields and the heart in CXR images. SCAN incorporates a critic network to impose on the convolutional segmentation network the structural regularities emerging from human physiology. During training, the critic network learns to discriminate between the ground truth organ annotations from the masks synthesized by the segmentation network. Through this adversarial process the critic network learns the higher order structures and guides the segmentation model to achieve realistic segmentation outcomes. Extensive experiments show that our method produces highly accurate and natural segmentation. Using only very limited training data available, our model reaches human-level performance without relying on any existing trained model or dataset. Our method also generalizes well to CXR images from a different patient population and disease profiles, surpassing the current state-of-the-art.