In multi-modal learning, some modalities are more influential than others, and their absence can have a significant impact on classification/segmentation accuracy. Hence, an important research question is if it is possible for trained multi-modal models to have high accuracy even when influential modalities are absent from the input data. In this paper, we propose a novel approach called Meta-learned Cross-modal Knowledge Distillation (MCKD) to address this research question. MCKD adaptively estimates the importance weight of each modality through a meta-learning process. These dynamically learned modality importance weights are used in a pairwise cross-modal knowledge distillation process to transfer the knowledge from the modalities with higher importance weight to the modalities with lower importance weight. This cross-modal knowledge distillation produces a highly accurate model even with the absence of influential modalities. Differently from previous methods in the field, our approach is designed to work in multiple tasks (e.g., segmentation and classification) with minimal adaptation. Experimental results on the Brain tumor Segmentation Dataset 2018 (BraTS2018) and the Audiovision-MNIST classification dataset demonstrate the superiority of MCKD over current state-of-the-art models. Particularly in BraTS2018, we achieve substantial improvements of 3.51\% for enhancing tumor, 2.19\% for tumor core, and 1.14\% for the whole tumor in terms of average segmentation Dice score.