Batch Normalization (BN) is one of the key components for accelerating network training, and has been widely adopted in the medical image analysis field. However, BN only calculates the global statistics at the batch level, and applies the same affine transformation uniformly across all spatial coordinates, which would suppress the image contrast of different semantic structures. In this paper, we propose to incorporate the semantic class information into normalization layers, so that the activations corresponding to different regions (i.e., classes) can be modulated differently. We thus develop a novel DualNorm-UNet, to concurrently incorporate both global image-level statistics and local region-wise statistics for network normalization. Specifically, the local statistics are integrated by adaptively modulating the activations along different class regions via the learned semantic masks in the normalization layer. Compared with existing methods, our approach exploits semantic knowledge at normalization and yields more discriminative features for robust segmentation results. More importantly, our network demonstrates superior ability in capturing domain-invariant information from multiple domains (institutions) of medical data. Extensive experiments show that our proposed DualNorm-UNet consistently improves the performance on various segmentation tasks, even in the face of more complex and variable data distributions. Code is available at https://github.com/lambert-x/ DualNorm-Unet.