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Unsupervised domain adaptation has increasingly gained interest in medical image computing, aiming to tackle the performance degradation of deep neural networks when being deployed to unseen data with heterogeneous characteristics. In this work, we present a novel unsupervised domain adaptation framework, named as Synergistic Image and Feature Alignment (SIFA), to effectively adapt a segmentation network to an unlabeled target domain. Our proposed SIFA conducts synergistic alignment of domains from both image and feature perspectives. In particular, we simultaneously transform the appearance of images across domains and enhance domain-invariance of the extracted features by leveraging adversarial learning in multiple aspects and with a deeply supervised mechanism. The feature encoder is shared between both adaptive perspectives to leverage their mutual benefits via end-to-end learning. We have extensively evaluated our method with cardiac substructure segmentation and abdominal multi-organ segmentation for bidirectional cross-modality adaptation between MRI and CT images. Experimental results on two different tasks demonstrate that our SIFA method is effective in improving segmentation performance on unlabeled target images, and outperforms the state-of-the-art domain adaptation approaches by a large margin.