Universal Domain Adaptation (UniDA) stands as a pivotal framework for transferring knowledge between disparate domains under the challenge of domain shifts. It uniquely caters to mismatches in class space without necessitating prior knowledge. Despite its utility, UniDA's effectiveness is often compromised in practical scenarios by the label shift, which poses significant hurdles to its generalization capabilities. Moreover, the existing UniDA approaches lack true universality, as they tend to erroneously detect "unknown"class samples even in situations where no "unknown"class are present, leading to unnecessary errors. In this paper, we propose a new framework known as Generalized Universal Domain Adaptation (GUDA) to handle the new challenges posed by both label shifts and the lack of true universality. At the core of our GUDA lies the Generalized Universal Adaptation Network (GUAN), comprising three innovative modules: the dual centroid learning module, which explores the internal structure of "unknown"classes and allows the model to determine if the "unknown"classes exist; the dual centroid employment module, which promotes cohesive clustering and emphasizes learning on minority classes to recognize "unknown"classes without prior knowledge; and the weighted multi-class adversarial alignment module, which aligns source and target samples with class- specific adjustments, preserving class boundaries amidst label shifts. These modules enable GUAN to form adaptable clusters tailored to varied target settings, enhancing classification precision and showcasing the true universality of our GUDA approach. Through extensive experimentation on benchmark datasets, we have demonstrated that our method substantially surpasses existing leading approaches in performance.