Brain extraction is a fundamental step for most brain imaging studies. In this paper, we investigate the problem of skull stripping and propose complementary segmentation networks (CompNets) to accurately extract the brain from T1-weighted MRI scans, for both normal and pathological brain images. The proposed networks are designed in the framework of encoder-decoder networks and have two pathways to learn features from both the brain tissue and its complementary part located outside of the brain. The complementary pathway extracts the features in the non-brain region and leads to a robust solution to brain extraction from MRIs with pathologies, which do not exist in our training dataset. We demonstrate the effectiveness of our networks by evaluating them on the OASIS dataset, resulting in the state of the art performance under the two-fold cross-validation setting. Moreover, the robustness of our networks is verified by testing on images with introduced pathologies and by showing its invariance to unseen brain pathologies. In addition, our complementary network design is general and can be extended to address other image segmentation problems with better generalization.