Video-text retrieval is an important but challenging research task in the multimedia community. In this paper, we address the challenge task of Unsupervised Domain Adaptation Video-text Retrieval (UDAVR), assuming that training (source) data and testing (target) data are from different domains. Previous approaches are mostly derived from classification based domain adaptation methods, which are neither multi-modal nor suitable for retrieval task. In addition, as to the pairwise misalignment issue in target domain, i.e., no pairwise annotations between target videos and texts, the existing method assumes that a video corresponds to a text. Yet we empirically find that in the real scene, one text usually corresponds to multiple videos and vice versa. To tackle this one-to-many issue, we propose a novel method named Uncertainty-aware Alignment Network (UAN). Specifically, we first introduce the multimodal mutual information module to balance the minimization of domain shift in a smooth manner. To tackle the multimodal uncertainties pairwise misalignment in target domain, we propose the Uncertainty-aware Alignment Mechanism (UAM) to fully exploit the semantic information of both modalities in target domain. Extensive experiments in the context of domain-adaptive video-text retrieval demonstrate that our proposed method consistently outperforms multiple baselines, showing a superior generalization ability for target data.