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Unsupervised domain adaptation enables intelligent models to transfer knowledge from a labeled source domain to a similar but unlabeled target domain. Recent study reveals that knowledge can be transferred from one source domain to another unknown target domain, called Universal Domain Adaptation (UDA). However, in the real-world application, there are often more than one source domain to be exploited for domain adaptation. In this paper, we formally propose a more general domain adaptation setting, universal multi-source domain adaptation (UMDA), where the label sets of multiple source domains can be different and the label set of target domain is completely unknown. The main challenges in UMDA are to identify the common label set between each source domain and target domain, and to keep the model scalable as the number of source domains increases. To address these challenges, we propose a universal multi-source adaptation network (UMAN) to solve the domain adaptation problem without increasing the complexity of the model in various UMDA settings. In UMAN, we estimate the reliability of each known class in the common label set via the prediction margin, which helps adversarial training to better align the distributions of multiple source domains and target domain in the common label set. Moreover, the theoretical guarantee for UMAN is also provided. Massive experimental results show that existing UDA and multi-source DA (MDA) methods cannot be directly applied to UMDA and the proposed UMAN achieves the state-of-the-art performance in various UMDA settings.