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Early Unsupervised Domain Adaptation (UDA) methods have mostly assumed the setting of a single source domain, where all the labeled source data come from the same distribution. However, in practice the labeled data can come from multiple source domains with different distributions. In such scenarios, the single source domain adaptation methods can fail due to the existence of domain shifts across different source domains and multi-source domain adaptation methods need to be designed. In this paper, we propose a novel multi-source domain adaptation method, Mutual Learning Network for Multiple Source Domain Adaptation (ML-MSDA). Under the framework of mutual learning, the proposed method pairs the target domain with each single source domain to train a conditional adversarial domain adaptation network as a branch network, while taking the pair of the combined multi-source domain and target domain to train a conditional adversarial adaptive network as the guidance network. The multiple branch networks are aligned with the guidance network to achieve mutual learning by enforcing JS-divergence regularization over their prediction probability distributions on the corresponding target data. We conduct extensive experiments on multiple multi-source domain adaptation benchmark datasets. The results show the proposed ML-MSDA method outperforms the comparison methods and achieves the state-of-the-art performance.