Though unsupervised domain adaptation (UDA) has achieved very impressive progress recently, it remains a great challenge due to missing target annotations and the rich discrepancy between source and target distributions. We propose Spectral UDA (SUDA), an effective and efficient UDA technique that works in the spectral space and can generalize across different visual recognition tasks. SUDA addresses the UDA challenges from two perspectives. First, it introduces a spectrum transformer (ST) that mitigates inter-domain discrepancies by enhancing domain-invariant spectra while suppressing domain-variant spectra of source and target samples simultaneously. Second, it introduces multi-view spectral learning that learns useful unsupervised representations by maximizing mutual information among multiple ST-generated spectral views of each target sample. Extensive experiments show that SUDA achieves superior accuracy consistently across different visual tasks in object detection, semantic segmentation and image classification. Additionally, SUDA also works with the transformer-based network and achieves state-of-the-art performance on object detection.