Abstract Unsupervised domain adaptation (UDA) conventionally assumes labeled source samples coming from a single underlying source distribution. Whereas in practical scenario, labeled data are typically collected from diverse sources. The multiple sources are different not only from the target but also from each other, thus, domain adaptater should not be modeled in the same way. Moreover, those sources may not completely share their categories, which further brings a new transfer challenge called category shift. In this paper, we propose a deep cocktail network (DCTN) to battle the domain and category shifts among multiple sources. Motivated by the theoretical results in [33], the target distribution can be represented as the weighted combination of source distributions, and, the multi-source UDA via DCTN is then performed as two alternating steps: i) It deploys multi-way adversarial learning to minimize the discrepancy between the target and each of the multiple source domains, which also obtains the source-specifific perplexity scores to denote the possibilities that a target sample belongs to different source domains. ii) The multi-source category classi- fifiers are integrated with the perplexity scores to classify target sample, and the pseudo-labeled target samples together with source samples are utilized to update the multi-source category classififier and the feature extractor. We evaluate DCTN in three domain adaptation benchmarks, which clearly demonstrate the superiority of our framework.