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Heterogeneous domain adaptation (HDA) tackles the learning of cross-domain samples with both different probability distributions and feature representations. Most of the existing HDA studies focus on the single-source scenario. In reality, however, it is not uncommon to obtain samples from multiple heterogeneous domains. In this article, we study the multisource HDA problem and propose a conditional weighting adversarial network (CWAN) to address it. The proposed CWAN adversarially learns a feature transformer, a label classifier, and a domain discriminator. To quantify the importance of different source domains, CWAN introduces a sophisticated conditional weighting scheme to calculate the weights of the source domains according to the conditional distribution divergence between the source and target domains. Different from existing weighting schemes, the proposed conditional weighting scheme not only weights the source domains but also implicitly aligns the conditional distributions during the optimization process. Experimental results clearly demonstrate that the proposed CWAN performs much better than several state-of-the-art methods on four real-world datasets.