Liver cancer is one of the most common cancers worldwide. Due to inconspicuous texture changes of liver tumor, contrast-enhanced computed tomography (CT) imaging is effective for the diagnosis of liver cancer. In this paper, we focus on improving automated liver tumor segmentation by integrating multi-modal CT images. To this end, we propose a novel mutual learning (ML) strategy for effective and robust multi-modal liver tumor segmentation. Different from existing multi-modal methods that fuse information from different modalities by a single model, with ML, an ensemble of modality-specific models learn collaboratively and teach each other to distill both the characteristics and the commonality between high-level representations of different modalities. The proposed ML not only enables the superiority for multi-modal learning but can also handle missing modalities by transferring knowledge from existing modalities to missing ones. Additionally, we present a modality-aware (MA) module, where the modality-specific models are interconnected and calibrated with attention weights for adaptive information exchange. The proposed modality-aware mutual learning (MAML) method achieves promising results for liver tumor segmentation on a large-scale clinical dataset. Moreover, we show the efficacy and robustness of MAML for handling missing modalities on both the liver tumor and public brain tumor (BRATS 2018) datasets. Our code is available at https://github.com/YaoZhang93/MAML.