Abstract: We introduce the problem of domain adaptation under Open Set Label Shift (OSLS), where the label distribution can change arbitrarily and a new class may arrive during deployment, but the class-conditional distributions $p(x|y)$ are domain-invariant. OSLS subsumes domain adaptation under label shift and Positive-Unlabeled (PU) learning. The learner's goals here are two-fold: (a) estimate the target label distribution, including the novel class; and (b) learn a target classifier. First, we establish the necessary and sufficient for identifying these quantities. Second, motivated by advances in label shift and PU learning, we propose practical methods for both tasks that leverage black-box predictors. Unlike typical Open Set Domain Adaptation (OSDA) problems, which tend to be ill-posed and amenable only to heuristics, OSLS offers a well-posed problem amenable to more principled machinery. Experiments across numerous semi-synthetic benchmarks on vision, language, and medical datasets demonstrate that our methods consistently outperform OSDA baselines, achieving $10$--$25\%$ improvements in target domain accuracy. Finally, we analyze the proposed methods, establishing finite-sample convergence to the true label marginal and convergence to optimal classifier for linear models in a Gaussian setup. Code is available at https://github.com/acmi-lab/Open-Set-Label-Shift.