Most unsupervised domain adaptation methods rely on rich prior knowledge about the source-target label set relationship, and they cannot recognize categories beyond the source classes, which limits their applicability in practical scenarios. This paper proposes a new paradigm for unsupervised domain adaptation, termed as Active Universal Domain Adaptation (AUDA), which removes all label set assumptions and aims for not only recognizing target samples from source classes but also inferring those from target-private classes by using active learning to annotate a small budget of target data. For AUDA, it is challenging to jointly adapt the model to the target domain and select informative target samples for annotations under a large domain gap and significant semantic shift. To address the problems, we propose an Active Universal Adaptation Network (AUAN). Specifically, we first introduce Adversarial and Diverse Curriculum Learning (ADCL), which progressively aligns source and target domains to classify whether target samples are from source classes. Then, we propose a Clustering Non-transferable Gradient Embedding (CNTGE) strategy, which utilizes the clues of transferability, diversity, and uncertainty to annotate target informative sample, making it possible to infer labels for target samples of target-private classes. Finally, we propose to jointly train ADCL and CNTGE with target supervision to promote domain adaptation and target-private class recognition. Extensive experiments demonstrate that the proposed AUDA model equipped with ADCL and CNTGE achieves significant results on four popular benchmarks.