Few-shot learning can adapt the classification model to new labels with only a few labeled examples. Previous studies mainly focus on the scenario of a single category label per example but have not solved the more challenging multi-label scenario with exponential-sized output space and low-data effectively. In this paper, we propose a semantic-aware meta-learning model for multi-label few-shot learning. Our approach can learn and infer the semantic correlation between unseen labels and historical labels to quickly adapt multi-label tasks from only a few examples. Specifically, features can be mapped into the semantic embedding space via label word vectors to explore and exploit the label correlation, and thus cope with the challenge on the overwhelming size of the output space. Then a novel semantic inference mechanism is designed for leveraging prior knowledge learned from historical labels, which will produce good generalization performance on new labels to alleviate the low-data problem. Finally, extensive empirical results show that the proposed method significantly outperforms the existing state-of-the-art methods on the multi-label few-shot learning tasks.